Exponents Power of k Problems
Posted on
27
Sep 2022

Exponents: Power of k Problems

Welcome to the penultimate article in our series on exponents on the GMAT/Executive Assessment (EA). Today we explore a problem category in which exponents are used to notating the highest power of an integer by which a larger integer is divisible. GMAT/EA problems typically use the variable k in the place of this exponent. These problems involve exponents but are also in the realm of number properties. Exponents are useful for notating the numbers of prime factors of a given integer. Here’s an example:

4680

468 * 10

234 * 2 * 2 * 5

117 * 2 * 2 * 2 * 5

13 * 9 * 2 * 2 * 2 * 5

13 * 3 * 3 * 2 *2 * 2 * 5

23 * 32 * 5 * 13

Since exponents notate successive multiplications by the same number, we can use them to consolidate lists of prime factors and immediately see how many of each prime factor exists within a number like 4680.

13 * 3 * 3 * 2 *2 * 2 * 5

23 * 32 * 5 * 13

In this example, 23 is a factor of 4680, but 24 is not a factor of 4680 because 4680 has only three prime factors of 2. 32 is a factor of 4680, but 33 is not a factor of 4680 because 4680 has only two prime factors of 3. A GMAT/EA problem might ask something like this: “What is the greatest integer k for which 2k is a factor of 4680?” The answer to this problem would be 3.

Let’s take a look at an official problem:

If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p?

(A) 10

(B) 12

(C) 14

(D) 16

(E) 18

This question effectively asks, “How many prime factors of 3 are in the number p?” The number p is “the product of the integers from 1 to 30 inclusive.” This is the verbalization of a piece of notation called the factorial. Factorials notate the multiplication of a positive integer by each positive integer less than itself. Here are some examples:

4! = 4 * 3 * 2 * 1

7! = 7 * 6 * 5 * 4 * 3 * 2 * 1

9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

To generalize the formula:

n! = n * (n – 1) * (n – 2) *  . . . * 1

Because the 1 at the end of the multiplication does not affect the value, factorials are often “spelled out” without the 1 at the end of the list.

Obviously, 30! Is too large of a number for us to evaluate. As a general rule, finding the actual value of any factorial larger than 7! or so is a waste of time. We are almost always more interested in the factors contained within a factorial.

30! = 30 * 29 * 28 * . . . * 1

So how many prime factors of 3 are contained within (30!)? Well, numbers like 29 and 28 don’t contain any prime factors of 3, because these numbers are not multiples of 3. The only factors of 30! that contain prime factors of 3 are the multiples of 3.

30 27 24 21 18 15 12 9 6 3

Some of these factors themselves contain more than one prime factor of 3. 27 is 33, so it has 3 threes. 9 is 32, so it has 2 threes. And a sneaky one is 18, which is not a power of 3 but is 2 * 9, or 2 * 32. So 18 also has 2 threes.

We can tally up all our prime factors of 3 like this:

30 27 24 21 18 15 12 9 6 3

1 3 1 1 2 1 1 2 1 1

If we add up the values in the second row, we obtain an answer of 14 prime factors of 3 in the number (30!). The correct answer is C.

This approach works, but there is actually a more straightforward way involving division. Consider first that performing the division a/b is effectively a way of counting how many multiples of b are less than or equal to a. 80 / 10 = 8, so 8 multiples of 10 are less than or equal to 80. 70 / 14 = 5, so 5 multiples of 14 are less than or equal to 70. When a is not perfectly divisible by b, the quotient still gives the number of multiples of b that are less than or equal to a, and the remainder is irrelevant. 56 / 9 = 6 remainder 2, so there are 6 multiples of 9 less than or equal to 56).

How does this help us identify prime factors of 3 in a number like (30!)? Well, if we take the quotient of 30/3, we can see how many of the factors of 30! are multiples of 3. In this case, there are 10. So there are at least 10 prime factors of 3 in the number (30!). If we take the quotient of 30/9 (or 30/(32), we see that 3 of these 10 multiples of 3 are also multiples of 9. Each of these numbers (9, 18, and 27) contains another prime factor of 3 that we did not “count” when we performed the division 30/3. Finally, we need to take 30/27 (or 30/33) to see that one of our 3 multiples of 9 is also a multiple of 27. The number, of course, is 27 itself, which contains a third prime factor of 3 that we didn’t “count” when we performed the divisions 30/3 and 30/9. 

quotient of 30 / 3 = 10

quotient of 30 / 9 = 3

quotient of 30 / 27 = 1

10 + 3 + 1 = 14 prime factors of 3 within 30!

To find the highest integer k for which nk is a factor of t!, add up the quotients of t/n, t/(n2), t/(n3) . . . for every power of n that is less than or equal to t.

 

Here’s one that requires some extra steps of algebra:

If n = 9! – 64, which of the following is the greatest integer k such that 3k is a factor of n?

(A) 1

(B) 3

(C) 4

(D) 6

(E) 8

This question effectively asks, “How many prime factors of 3 are in the number 9! – 64?

Let’s start by viewing 9! and 64 in factored forms:

9! – 64

(9 * 8 * 7 * 6 * 5 * 4 * 3 * 2) – (6 * 6 * 6 * 6)

We can go one step further and break all the non-primes down to primes:

(9 * 8 * 7 * 6 * 5 * 4 * 3 * 2) – (6 * 6 * 6 * 6)

(3 * 3) * 23 * 7 * (2 * 3) * 5 * 22 * 3 * 2 – (24 * 34)

Since each factor of 6 in 64 is equal to (3 * 2), we can represent the group of 4 sixes as (24 * 34). Now to consolidate the rest:

(3 * 3) * 23 * 7 * (2 * 3) * 5 * 22 * 3 * 2 – (24 * 34)

(27 * 34 * 5 * 7) – (24 * 34)

The subtraction is preventing us from seeing how many threes are in the integer version of this number. We will have to extract a factor of (24 * 34), like this:

(27 * 34 * 5 * 7) – (24 * 34)

(24 * 34)[(23 * 5 * 7) – 1]

Now we can evaluate the bracketed expression in order to prime factorize again:

(24 * 34)[(23 * 5 * 7) – 1]

(24 * 34)[(8 * 5 * 7) – 1]

(24 * 34)[280 – 1]

(24 * 34)(279)

And we’ll break 279 down into prime factors:

(24 * 34)(279)

(24 * 34)(9 * 31)

(24 * 34)(32 * 31)

24 * 36 * 31

Finally we can see that there are 6 prime factors of 3 in the number 9! – 64, and the correct answer is D.

Here’s a final problem for today:

For the positive integers a, b, and k, ak|  |b means that ak is a divisor of b but a(k+1) is not a divisor of b. If k is a positive integer and 2k|  |72, then k is equal to

(A) 2

(B) 3

(C) 4

(E) 8

(D) 18

This problem uses a random symbol – two vertical lines – and then defines it for you. This is common practice on GMAT/EA problems and is simply a way to “disguise” the type of problem you’re looking at. Think about the way they’ve defined this symbol, and you’ll see that this problem is asking the same thing as our other examples: what is the greatest integer k for which 2k is a factor of 72? Or to put it even more simply: How many prime factors of 2 does 72 have?

72 = 8 * 9 = 23 * 32

There are 3 prime factors of 2 in 72, so the correct answer is B.

This concludes our brief study of the “power of k” problems on the GMAT/EA. Join us next time for the final article in our series on exponents to learn how to handle problems with impossibly large numbers like 287459.

If you are in the middle of studying for the GMAT/EA and are looking for a private GMAT tutor, our elite tutors have all scored over 770 on the GMAT and have years of professional experience with tutoring. You can meet with us for a 30-minute complimentary consultation call.

Contributor: Elijah Mize (Apex GMAT Instructor)

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Zeros and Nonzeros
Posted on
21
Sep 2022

Zeros and Nonzeros

Welcome back to our series on exponents. Last time we used powers of 10 to express “almost integer” numbers. Today we will use powers of 10 to handle problems that ask us to count zeros or nonzero digits. These problems can be baffling if you haven’t learned about them. Let’s start by comparing two official GMAT/Executive Assessment (EA) problems in this category:

If t = 1/(29+53) is expressed as a terminating decimal, how many zeros will t have between the decimal point and the first nonzero digit to the right of the decimal point?

A. Three

B. Four

C. Five

D. Six

E. Nine

If d = 1/(23+57) is expressed as a terminating decimal, how many nonzero digits will d have?

A. One

B. Two

C. Three

D. Seven

E. Ten

In each of these problems, we see the number 1 divided by powers of 2 and powers of 5. The first problem asks us to count the number of zeros between the decimal and the first nonzero digit. The second problem asks us to count the nonzero digits instead.

Whether counting zeros or nonzeros, the best way to start on these problems is to “extract” the powers of 10 in the denominator. Let’s take the first problem:

1 / (29 * 53)

Recall our fundamental fact about exponents: they notate successive multiplications by the same value. So this denominator is the product of nine twos and three fives. We can make powers of 10 by pairing two with fives (since 2 * 5 = 10).

1 / (29 * 53)

1 / 26 * (23 * 53)

1 / 26 * 103

Three of our twos went over and joined the fives to make three tens, or 103.

Now we have 1 / (26 * 103), and we need to see how many zeros appear after the decimal but before any nonzero digits. At this point, knowing your powers of 2 comes in handy and allows you to find the full value of the denominator.

1 / 26 * 103

1 / 64 * 103

So we have 1 / 64,000. We can find the number of zeros in the decimal form of this number by simply subtracting 1 from the number of digits in the denominator. The denominator, 64,000, has five digits, so 1 / 64,000 has four zeros between the decimal and the first nonzero digit. The correct answer is B.

Let’s look again at our second example problem:

If d = 1/(23+57) is expressed as a terminating decimal, how many nonzero digits will d have?

A. One

B. Two

C. Three

D. Seven

E. Ten

This problem asks us about the nonzeros instead of the zeros. Since every digit is either a nonzero or a nonzero, we can find the number of nonzero digits most easily by subtracting the number of zeros after the decimal from the total number of digits after the decimal.

# of nonzero digits after decimal = (# of digits after decimal) – (# of zeros after the decimal)

We know how to find the number of zeros after the decimal, but first, we will find the total number of digits after the decimal. On these problems, this number is always equal to the larger exponent in the base. Here the larger exponent is 7, so there are a total of 7 digits after the decimal.

Now, all we have to do is find the number of zeros before the first nonzero digit (like we did on the previous problem) and subtract this number from 7.

1 / (23 * 57)

1 / (54 * 103)

Again, knowing your powers helps:

1 / (54 * 103)

1 / (625 * 1000)

1 / (625,000)

The denominator has six digits, so there are 6 – 1 = 5 zeros after the decimal before the first nonzero digit. There are a total of seven digits after the decimal.

# of nonzero digits after decimal = (# of digits after decimal) – (# of zeros after the decimal)

# of nonzero digits after decimal = 7 – 5

# of nonzero digits after decimal = 2

And the correct answer choice is B.

Now you’re ready for “zero or nonzero” GMAT/EA problems. Next time we will look at a problem category that merges exponents with number properties.

Contributor: Elijah Mize (Apex GMAT Instructor)

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“Almost an Integer” Problems
Posted on
20
Sep 2022

“Almost an Integer” Problems

Unless you do the math as a career or a hobby, you probably prefer integers to non-integers. Whole numbers are easier for us to conceptualize. But a certain class of GMAT/Executive Assessment (EA) problems involves numbers that are almost integers. Generally, this nearest integer is 1, so the nearby numbers look like this:

0.99999

0.9995

1.001

1.000006

Whenever you see a number like one of these on a GMAT/EA problem, you should use powers of 10 to notate the difference between the value in question and the nearest integer.

Let’s start by expressing our first example number, 0.99999, as a difference from 1 without powers of 10. Then we’ll convert that difference to a power of 10.

0.99999 = 1 – 0.00001

When the number you’re working with is of the form 0.999 . . . , the difference from 1 is all zeros with a 1 at the end, and the number of digits after the decimal remains consistent. Here there are five nines after the decimal, so our difference from 1 has four zeros and then a 1 at the end for a total of five digits after the decimal.

Now to convert our difference from 1 to scientific notation. The number in question, 0.00001, is small, so the power of 10 will be negative. The rule is to simply use the negative version of the number of digits after the decimal. According to this rule, 0.00001 = 10-5. Therefore we have this:

0.99999 = 1 – 0.00001 = 1 – 10-5

Let’s try the next example from above: 0.9995. This time there are only four digits after the decimal, but the last one is a 5 instead of a 9. Again, let’s express this value as a difference from 1 without scientific notation first:

0.9995 = 1 – 0.0005

As before, the number of digits after the decimal must remain consistent. But instead of using all zeros and then a single 1, we use all zeros and then whatever digit sums to 10 with the final digit of the original number. When the final digit of the original number is a 9, as in the first example, we use a 1 (since 9 + 1 = 10). In this case, the final digit of the original number is 5, so we need to use another 5 (5 + 5 = 10) to finish off our difference from 1.

0.9995 = 1 – 0.0005 = 1 – 5*10-4

There are four digits after the decimal, so the exponent of the 10 is -4. The coefficient of 5 is applied to the 10-4 term because 0.0005 = 5 * 0.0001 = 5*10-4.

We can solidify this with a general rule for finding decimal differences from 1. The number of digits after the decimal must remain consistent, and the digits in each place must sum to 9, except for the final digits which sum to 10. Here’s an example:

0.8653 = 1 – 0.1347

Here are the sums of the tenths, hundredths, thousandths, and ten-thousandths digits:

8 + 1 = 9

6 + 3 = 9

5 + 4 = 9

3 + 7 = 10

Here are some numbers you can use for practice. Their differences from 1 are at the end of the article.

0.23468

0.9834

0.31479

0.34098

0.999357

0.00042

0.000257

This covers numbers slightly less than 1. Numbers slightly greater than 1, like the examples from before of 1.001 and 1.000006, are easier to work with because you can convert everything after the decimal directly to a power of 10 without having to find a difference from 1.

1.001 = 1 + 10-3

1.000006 = 1 + 6*10-6

With these skills in place, you’re ready to tackle some official problems.

(1.00001)(0.99999) – (1.00002)(0.99998) =

(A) 0

(B) 10-10

(C) 3(10-10)

(D) 10-5

(E) 3(10-5)

Let’s convert each of the four numbers in the problem:

1.00001 = 1 + 10-5

0.99999 = 1 – 10-5

1.00002 = 1 + 2*10-5

0.99998 = 1 – 2*10-5

And we have a lovely pattern emerging.

(1 + 10-5)(1 – 10-5) – (1 + 2*10-5)(1 – 2*10-5)

Now all that remains is to “foil” the expressions and then simplify:

(1 + 10-5)(1 – 10-5) – (1 + 2*10-5)(1 – 2*10-5)

(1 – 10-5 + 10-5 – 10-10) – (1 – 2*10-5 + 2*10-5 – 4*10-10)

Now all the 10-5 terms cancel:

(1 – 10-10) – (1 – 4*10-10)

Now the 1s cancel as well:

-(10-10) + 4(10-10)

3(10-10)

And the correct answer is C.

Here’s another:

1 – 0.00001 = 

(A) (1.01)(0.99)

(B) (1.11)(0.99)

(C) (1.001)(0.999)

(D) (1.111)(0.999)

(E) (1.0101)(0.0909)

This one is different because the first step has already been done for us. Instead of starting with 0.999999, the problem starts in “difference from 1” form. All we have to do is convert the difference to a power of 10:

0.000001 = 10-6

1 – 0.000001 = 1 – 10-6

Now what to make of the answer choices? After a quick scan, the only ones that look very friendly to a “1 +/- 10-x” form are A and C. Choice A can’t be right because the factors 1.01 and 0.99 contain a total of only four digits after their decimals, and the product we are looking for, 1 – 10-6 or 0.999999, has six digits after the decimal. In multiplication, the number of digits after the decimal in the product always matches the total number of digits after the decimals in the factors.

So answer choice C looks like the best candidate. Let’s convert it to “power of 10” form:

(1.001)(0.999) = (1 + 10-3)(1 – 10-3)

Now we can “foil” the expression and simplify:

(1 + 10-3)(1 – 10-3) = 1 + 103 – 103 – 10-6 = 1 – 10-6

And as we suspected, answer choice C turned out to be correct.

Now you’re ready to handle “almost integers” on GMAT/EA problems. Next time we’ll use powers of 10 to address problems that ask about zeros or nonzero digits.

If you are looking for professional help to boost your GMAT/EA performance, head to our official website and book your 30 minutes complimentary assessment session now with our top tutors.

Solutions to drills:

0.23468 = 1 – 0.76532

0.9834 = 1 – 0.0166

0.31479 = 1 – 0.68521

0.34098 = 1 – 65912

0.999357 = 1 – 0.000643

0.00042 = 1 – 0.99958

0.000257 = 1 – 0.999743

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Exponent Properties in Data Sufficiency
Posted on
08
Sep 2022

Exponent Properties in Data Sufficiency

Exponents have properties that make them ideal pieces for data sufficiency problems on the GMAT and Executive Assessment (EA) quant sections. We’ve surveyed most of these properties in our first four articles in this series, but a couple of key ones haven’t been mentioned explicitly.

First, x0 = 1. A full mathematical explanation of this property is beyond the scope of this series of articles and is unnecessary for GMAT/EA preparation. If you know the rule, you can employ it as needed. The debated exception 00 exists but does not occur on GMAT quant.

Second, the numbers 0 and 1 are both “immune” to exponents. We mentioned this rule for 1 in our last article, but it is important to know that the rule applies to 0 as well. 0x = 0, and 1x = 1.

Another important rule for exponent DS problems has already been mentioned: even powers of negative numbers are positive, and odd powers of negative numbers are odd. To extend this, xeven is always greater than or equal to 0 (it is only equal to 0 when x = 0; in all other cases xeven is positive).

A basic rule to remember is that positive numbers have two square roots which are negative and positive versions of the same value. If b2 = 16, b may equal 4 or -4. Forgetting about the possibility of the negative square root usually leads to incorrect answers.

One more important rule to remember is that if x is positive, then xy is positive. No exponent can cause an exponential expression with a positive base to have a negative value or a value of 0. To state the same rule differently, exponential expressions with negative values have negative bases.

Let’s get into some official problems:

Is zp negative?

  1. pz4 < 0
  2. p + z4 = 14

We will need to know the signs of the variables p and z. If they are both positive or both negative, then the product zp is positive. If z or p is negative and the other positive, then the product zp is negative.

Statement 1 tells us that the product p * z4 is negative. Therefore either of p or z4 is negative and the other is positive. Since the exponent 4 is positive, z4 is always positive (or 0, but the statement rules out that possibility). This means that p must be negative. This isn’t enough to answer whether zp is negative, but it may be useful if we have to combine statements 1 and 2.

Statement 2 tells us that the sum of p and z4 equals 14. There are simply too many possibilities for this statement on its own to be sufficient.

Now we must combine statements 1 and 2. We know from statement 1 that p is negative. Therefore z4 must be positive in order for p + z4 to have the positive sum of 14. But we already know that z4 is positive because the exponent 4 is even! A common mistake in these DS problems is to accidentally transfer information about z4 back to the variable z itself. Doing so on this problem might lead you to incorrect answer choice C. In this case, statements 1 and 2 together are still not sufficient, and the correct answer is E.

Let’s try another:

If r, s and t are nonzero integers is r5s3t4 negative?

  • rt is negative.
  • s is negative.

Here’s another even exponent of 4. This means that t4 is either positive, or 0 in the case of t = 0. If t = 0, then the product r5s3t4 is 0 and not negative.

Statement 1 tells us that the product rt is negative. Therefore either of r or t is negative and the other is positive. We still know nothing about the variable s, so this can’t be sufficient on its own.

Statement 2 tells us that s, and therefore s3, is negative, but it says nothing about r or t. This can’t be sufficient on its own.

Combining the statements, we can approach statement 1 with two pieces of info: that t4 is positive (or 0) and that s3 is negative (since s is negative). Therefore the product r5s3t4 is (?) * (-) * (+), and the sign of the variable r makes all the difference.

Returning to statement 1, we consider again that the product rt is negative. This means that either r or t is negative, but unfortunately, we don’t know which one. Again, if we mistakenly jump from the fact “t4 is positive” to the unfounded conclusion “t is positive,” we will select incorrect answer choice C. Again, the statements together are insufficient, and the correct answer is E.

Here’s another:

If a and b are integers is a5< 4b?

  1. a3= -27
  2. b2= 16

The upshot of statement 1 is that a is negative. Therefore a5 is also negative, since 5 is another negative exponent. To get specific (which is probably unnecessary for the problem), a = -3 and a5 = -243.

We don’t know anything about b, so it’s tempting to conclude that statement 1 alone is insufficient. But we are asked to compare a5 against 4b, not against b itself. If you remember the rule that exponential expressions with positive bases have positive values, you’ll see that statement 1 alone is sufficient. a5 is negative, and 4b is positive.

Statement 2 tells us that b = 4 or -4. Therefore 4b = 256 or 1/256. Either way, without statement 1, we know nothing about a5. Statement 2 on its own is insufficient, and the correct answer is A.

Here’s a final DS exponents problem:

If x and y are integers, is x > y?

  1. x + y > 0
  2. yx < 0

Statement 1 is certainly insufficient by itself, but we should still think about what it tells us. Either x and y are both positive, or one is negative and the other is positive, with the positive number having the greater absolute value.

Statement 2 tells us that yx is negative. The rule we need is that exponential expressions with negative values have negative bases. This means y is negative.  On its own, this statement is still insufficient. But combined with statement 1, we know that x must be greater than y, because x must be positive in order to produce a positive sum with a negative number y. The statements together are sufficient, and the correct answer is C. 

We are halfway through our series on exponents and have covered all the basics. The remaining five articles will cover specific problem types involving exponents.

If you are in the middle of studying for the GMAT and are looking for a private GMAT tutor, our elite tutors have all scored over 770 on the GMAT and have years of professional experience with tutoring. You can meet with us for a 30-minute complimentary consultation call.

Contributor: Elijah Mize (Apex GMAT Instructor)

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Posted on
07
Sep 2022

Bases Between -1 and 1

Many GMAT and Executive Assessment (EA) exponent problems – especially data sufficiency ones – require you to consider fractional bases. By this I mean proper fractions with values between -1 and 1, not improper fractions whose numerators exceed their denominators.

There are four “kinds” of bases separated by the three “boundary points” of -1, 0, and 1. Numbers in each of the four “zones” separated by these values behave similarly as bases of exponents. On many DS problems, we need to consider numbers less than -1, negative fractions, positive fractions, and numbers greater than 1, as well as the boundary points of -1, 0, and 1. This sounds like a lot of work, but practice will build your “spidey sense” for when a certain kind of base leads to an exception and an insufficient statement.

Let’s handle the “boundary points” first. As mentioned in a prior article, the boundary point of 1 is simple because it is “immune” to exponents: 1anything = 1. Likewise, 0positive = 0. For -1, (-1)even = 1 and (-1)odd = -1. Since (-1)anything equals either 1 or -1, it is worth noting that the absolute value of -1anything is 1. If you are unfamiliar with absolute value, don’t worry, it’s a simple concept: absolute value is a number’s distance from 0 on a number line. The absolute value of a positive number is . . . itself. The absolute value of a negative number is simply the positive version of the number. For positive numbers, “normal” value and absolute value are the same and trend together. For negative numbers, absolute value increases as “normal” value decreases. Absolute value is notated with vertical lines on either side of a value, variable, or expression. |(-1)anything| = 1.

Now for the four ranges of numbers. Numbers greater than 1 are the simplest. For these numbers, the higher the exponent, the higher the overall value. For numbers greater than 1, higher powers have higher values.  Numbers less than -1 are only slightly more complex. For numbers less than -1, higher powers have higher absolute values, but odd powers are negative and even powers are positive.

Now for the positive and negative fractions: the more times you multiply a fraction by itself, the closer the resulting value gets to 0. (¾)2 or ¾ * ¾, which we can read as “¾ of ¾,” is less than ¾. For positive fractions, higher powers have lower values. For negative fractions, higher powers have lower absolute values, but odd powers are negative and even powers are positive.

Let’s demonstrate all of our rules with the examples of 2, -2, ½, and -½.

2 < 22 < 23 . . .

|2| < |22| < |23| . . .

 

(-2)5 < (-2)3 < (-2) < 0 < 22 < 24 < 26 . . .

|-2| < |(-2)2| < |(-2)3| . . .

 

½ > (½)2 > (½)3 . . .

|½| > |(½)2| > |(½)3| . . .

 

-½ < (-½)3 < (-½)5 < 0 < (-½)6 < (-½)4 < (-½)2

|-½| > |(-½)2| > |(-½)3| . . .

The patterns for the negatives can take some getting used to, so study these rules frequently and, more importantly, build your fluency with practice problems! Here’s a straightforward one:

Is x2 greater than?

  1. x2 is greater than 1.
  2. x is greater than -1.

For statement 1, it helps to remember that only numbers greater than 1 or less than -1 can have powers greater than 1. Powers of fractions are always fractions. So if x2 is greater than 1, x is either greater than 1 or less than -1. If x is less than -1, then x2, which according to the statement is greater than 1, is greater than x. And if x is greater than 1, it still gets larger when it is squared, so x2 is always greater than x, and statement 1 is sufficient.

Statement 2 tells us that x is greater than -1. If we remember our boundary points, we can solve this one without having to think about fractions. X could be 0 or 1, and in either case, x2 is equal to, not greater than, x. But for any number greater than 1, x2 is greater than x. So statement 2 on its own is insufficient, and the correct answer is A.

Let’s try another DS problem:

Is xy > x2y2?

  1. 0 < x2 < 1/4
  2. 0 <  y2 < 1/9

To verbalize the question: is the absolute value of the product xy greater than the square of the product xy?

Statement 1 tells us that x2 is a positive fraction, which means that x itself is a fraction with a greater absolute value, but we don’t know whether it is positive or negative. Without knowing anything about y, this isn’t enough. Statement 1 alone is insufficient. Statement 2 is similar and also insufficient. Taking the statements together, we know that the absolute value of x is greater than x2, and the absolute value of y is greater than y2. It follows that the absolute value of the product xy is greater than x2y2. We don’t know whether x and y are positive or negative, but we’re talking absolute value so it doesn’t matter. The statements together are sufficient, and the correct answer is C.

Finally, let’s see what happens when fractional bases meet negative exponents:

(1/2)-3(1/4)-2(1/16)-1=

A. (1/2)(-48)

B. (1/2)(-11)

C. (1/2)(-6)

D. (1/8)(-11)

E. (1/8)(-6)

This problem benefits from “translating” out of the negative-exponent form. (½)-3 = 23, (¼)-2 = 42, and (1/16)-1 = 16. Ideally, you recognize that everything in the expression can be converted to powers of 2. 42 and 16 both equal 24, so the full expression is 23 * 24 * 24, or, remembering your exponent rules, 211. If you noticed how the fractional base and the negative exponent “canceled” each other, you should recognize 211 as (½)-11, answer choice B.

If you are looking for extra help in preparing for the GMAT, we offer extensive one-on-one GMAT tutoring for high-achieving students. You can schedule a complimentary, 30-minute consultation call with one of our tutors to learn more! 

Contributor: Elijah Mize (Apex GMAT Instructor)

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Negative Exponents and Negative Bases_
Posted on
30
Aug 2022

Negative Exponents and Negative Bases

Welcome back to our series on exponents. Today we will see what happens when we throw negatives into our exponential expressions. We will explore both negative bases and negative exponents.

First, the bases. The rule to remember for negative bases is that odd powers of negative bases are negative and even powers of negative bases are positive. This rule makes sense when you remember that exponents simply notate a number of multiplications by the base (and remember your rules about multiplication with negative factors). 

Multiplication with an odd number of negative factors yields a negative product: 

(-x)3 = -x * -x * -x

(-x)3 < 0

Multiplication with an even number of negative factors yields a positive product:

(-x)4 = -x * -x * -x * -x

(-x)4 > 0

A note on notation: parentheses should always be used around a negative value as the base of an exponent. If they are not, then the order of operations dictates that the exponent be applied before the negative sign. To avoid confusion, whenever the negative is meant to be left out of the exponential operation, parentheses are used like this: -(x)4 to make the order clear. Please note that -(x)4 is less than 0, and (-x)4 is greater than 0.

Now for negative exponents. Here is a simple rule is the best way to explain it:

x-n = 1/xn

A negative exponent indicates a value reciprocal to the value with a positive exponent. It’s good practice to “translate” any exponential expressions with negative exponents to their reciprocal positive forms. Seeing it both ways can help you make sense of problems.

5-4 = 1 / (54) = 1 / 625

17-2 = 1 / (172) = 1 / 289

(9 / 16)-2 = (16 / 9)2 = 256 / 81

Integer bases with negative exponents go under a numerator of 1; fractional bases with negative exponents simply flip. Let’s look at some “double negative” exponential expressions.

(-6)-3 = 1 / (-6)3 = 1 / -216

(-2)-10 = 1 / (210) = 1 / 1024

(-4 / 3)-4 = (-¾)4 = 81 / 256

Let’s get into some official GMAT problems. Be careful with this first one!

From the consecutive integers -10 to 10 inclusive, 20 integers are randomly chosen with repetitions allowed. What is the least possible value of the product of the 20 integers?

  1. (-10)20
  2. (-10)10
  3. 0
  4. -(10)19
  5. -(10)20

One incorrect answer is chosen far more often than any other on this problem: answer choice D. Trying to minimize the product, many people consider taking the maximum number (20) of the lowest value (-10). The common mistake is then thinking that -(10)20 (answer choice E) is actually a positive value since it involves an even number (20) of negative factors. Many people then take “the next best thing” in answer choice D, which shifts the exponent to the next odd number down from 20.

In fact, -(10)20 does not involve an even number of negative factors, since the negative sign is excluded from the exponential expression by the parentheses. Answer E means “take 1020 and make it negative.” It is true that taking 20 negative tens and multiplying them all together produces a large positive value (the opposite of what we are aiming for on this problem), but this misguided idea is notated by answer choice A – not answer choice E. Remember that the answer choices notate the product of the 20 factors, not necessarily a condensed list of the 20 factors. 

It is possible to choose 20 integers from -10 to 10 inclusive that, when multiplied, yield a product of -(10)20 (answer choice E). The “least possible value” is obtained by finding the greatest absolute value (distance from 0) in negative form. So we want all 20 of our factors to be either 10 or -10 since this will maximize the absolute value (distance from 0) of the product. To ensure that the product is also negative, we simply need an odd number of negative tens. We can use nineteen negative 10s and 1 positive 10, 1 negative 10 and 19 positive 10s, or any odd combination in between. Any of these options will yield a product of -(10)20. Read the notation carefully!

Let’s try another:

The value of 2(-14)+2(-15)+2(-16)+2(-17)/5 is how many time the value of 2(-17)?

  1. 3/2
  2. 5/2
  3. 3
  4. 4
  5. 5

This problem benefits from the skill of noticing patterns and “checking” them. You should see the pattern in the numerator and generalize it by saying “the negative exponent on the 2 keeps decreasing by 1.” Then you can see how this pattern “works” by checking a single case. 

2-17 = 1 / 217

2-16 = 1 / 216

Since 217 = 2 * 216, (1 / 217) is half the value of (1 / 216). Or, to say it a more useful way, 2-16 = 2 * 2-17 This pattern will continue through the numerator. Since we are looking for how many copies of 2-17 we have in this expression, we can replace 2-17 with 1 and follow the pattern.

(214 + 2-15 + 2-16 + 2-17) / 3

(8 + 4 + 2 + 1) / 3

15 / 3 = 3

And the correct answer is C.

Here’s a final problem for today:

a is a nonzero integer. Is 

a2greater than 1?

  • a < -1
  • a is even.

To evaluate statement 1, simply start by checking a = -2. (-2)-2 = 1 / (-2)2  = ¼. Moving on to a = -3, (-3)-3 = 1 / (-3)3 = 1 / -27. This time the value is negative, but the positive value from a = -2 is still less than 1. If you imagine continuing with a = -4, a = -5, etc., you will just keep making smaller and smaller fractions. Statement 1 alone is sufficient.

Statement 2 on its own is easy to check since we already know from checking statement 1 that some even values for the variable a yield an aa with a value less than 1. And it shouldn’t be hard to imagine an even value for variable a where aa is greater than 1. For example, 22 = 4 and 44 = 256. So statement 2 alone is insufficient, and the correct answer is A.

If you went with answer choice C, here’s what might have happened. Noticing that statement 1 tells you the base is negative, you might have seen next that statement 2 tells you the exponent is even. You might have thought that this “evenness” of the exponent makes the difference since it determines the positivity or negativity of the expression. Very often in DS problems with negative bases, the even/odd identity of the exponent really matters. But in this case, it’s a trap, because we were asked whether aa is greater than 1 (not 0), and the fact that the exponent is also negative means that it’s even/odd identity is irrelevant – the value is always less than 1.

The rules governing negative exponents and negative bases are simple, but the GMAT and EA problems that employ these rules can catch you if you aren’t careful. Next time we will look at another tricky exponential scenario: when the base is between -1 and 1.

Register now for a free consultation with one of our top tutors.

Contributor: Elijah Mize (Apex GMAT Instructor)

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Posted on
24
Aug 2022

Undoing Exponents: Radicals and Roots

As you will definitively have to deal with radicals on the GMAT and Executive Assessment (EA), we’ve put together an article for you to master the topic. Addition is “undone” by subtraction, multiplication is “undone” by division, and the powers notated by exponents are “undone” with a piece of notation called a radical.

√  the radical

The number, variable, or expression inside or under the radical is referred to as the radicand, and sometimes there is a small number called the index nestled in the “crook” of the radical.

When no index appears, the index is understood to have a value of 2. (This will make sense momentarily.) Unfortunately there exists no one-word name – like “addition” or “multiplication” for what exponents do. For an exponent of n, we use the phrase “raising to the nth power.” For a radical with an index of n, we use the phrase “taking the nᵗʰ root.” Exponents notate powers, radicals notate roots. If there is an “invisible” index of 2, the notated value is called the square root of the radicand (the number inside or under the radical). If the index is 3, the notated value is called the cube root of the radicand. For all integers above 3, we use the usual ordinals (fourth root, fifth root, etc.)

√36 the square root of 36

³√125 the cube root of 125

√81 the fourth root of 81

If you’ve studied your powers chart from the first article in this series, you should recognize those radicands. Here’s a simple way to represent how radicals “undo” exponents:

x = a     √a = x

If the nᵗʰ power of x equals a, then the nth root of a equals x. Radical expressions like this notate the number x which, when multiplied by itself n times (or, to use exponents, raised to the nᵗʰ power), equals a. Exponential expressions “start with” a base and raise it to a power, notating a value (in our case the variable a). Radical expressions “start with” the full value notated by some exponential expression and use the exponent from this expression as a root, notating the base of the exponential expression (in our case the variable x). To use our examples above:

√36  the square root of 36  x² = 36, x = 6

³√125  the cube root of 125  x³ = 125, x = 5

√81  the fourth root of 81  x = 81, x = 3

Now you have another reason to learn your common powers: when you see one of those special numbers underneath a radical, you can quickly evaluate the radical expression. This is akin to how knowing your times tables makes division really easy.

Another quality of roots is that they can be “translated” into fractional exponents according to the following rule:

√(x) = xᵃ⁄ⁿ

(√x) = xᵃ⁄ⁿ

Two forms are shown because the position of the exponent a is irrelevant. It can be considered “within” and “before” the radical operation as in the first version or “above” and “after” the radical operation as in the second version.

As always, this rule can be reversed.

xᵃ⁄ⁿ = n√(xa)

xᵃ⁄ⁿ = (n√x)a

Seeing fractional exponents written in GMAT and EA questions is quite rare, but sometimes a given radical expression that doesn’t break down easily is better notated as a fractional exponent for the sake of seeing potential algebraic simplifications.

This correlation between radicals and fractional exponents brings up the key point that radicals and exponents are just different – or reverse – ways of notating the same thing. Therefore all the exponent rules apply to radicals as well. The radical versions appear less frequently than the exponent versions but are still valuable pieces in your toolkit.

Equal bases with different roots are rather unusual; combining or collapsing roots with the same index is certainly more common. And just as we can have a “power of a power,” we can have a “root of a root.”

These rules bring up a final important point about radical expressions: they often need to be simplified. Mathematicians don’t like to leave numbers inside radicals when it is possible to express the same value differently, so many answer choices on the GMAT and EA involve simplified versions of radical expressions. Here’s an example for our purposes:

√630

We’ll have to get into some number properties to show what happens here. One of our radicals rules states that this radicand 630 can be broken into factors and represented as multiple radical expressions. Let’s break down 630 to its prime factors.

630 = 2 * 3² * 5 * 7

√630 = √(2 * 3² * 5 * 7)

Given an index of n, any n (number) of the same prime factor can be placed inside their own radical. Since we are taking a square root here (with an understood index of 2), we only need two of the same prime factor in order to do this. Here we can do it with our 3s.

√630 = √(2 * 3² * 5 * 7)

√630 = √(3²) * √(2 * 5 * 7)

Now the power of 2 and the root of 2 on the 3 cancel each other out.

√630 = √(32) * √(2 * 5 * 7)

√630 = 3 * √(2 * 5 * 7)

√630 = 3√70

When simplifying a radical expression, use the index as a key. Given an index of n, look for sets of n (number) of the same prime factor. For each of these sets, bring one of that prime factor out in front of the radical. Here’s another example:

³√8640

A key step to shortening the process is recognizing 216 as 63. Let’s line up our prime factors:

2 * 3³ * 5

Remember that we’re taking a cube root this time, so we’re looking for sets of three of the same prime factor. We can make two sets from our 2s (since we have six of them) and one set from our 3s. So two 2s and one 3 come out from the radical, leaving only the 5.

³√8640 = 2 * 2 * 3 * ³√5 = 12³√5

To close this article, let’s try a couple of official GMAT problems involving radicals:

If n = √(16/81) , What is the value of √n?

  1. 1/9
  2. 1/4
  3. 4/9
  4. 2/3
  5. 9/2

This problem benefits from knowing both your rules and your powers. Knowing your rules lets you represent this radical expression as √16/√81, and knowing your powers lets you easily recognize √16 as 4 and √81 as 9, leading to answer choice C, 4/9. Here’s one more:

In the formula w= P/√v, Integers p and t are positive constants. If w=2 when v=1 and if w=1/2 when v=64, then t=

  1. 1
  2. 2
  3. 3
  4. 4
  5. 16

First we are told that w = 2 when v = 1. 

2 = P / √1

Here you need to recognize a fact we haven’t mentioned yet: that any root (or power) of 1 is still 1. This makes sense when you think about multiplying 1 by itself again and again – the value never changes. Therefore the denominator of the expression equals 1, and P must equal 2. Armed with this knowledge, you can make use of the second given fact: w = ½ when v = 64.

½ = 2/√64

4 = √64

If you recognize 64 as 26, you can use your “power to a power” rule to change 26 to 43, leading you to answer choice C.

This concludes our survey of radicals on the GMAT and EA. Next time we will introduce negatives to exponential expressions.

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Posted on
13
Jul 2022

Enhance Your GMAT Strategy in Under 20 Minutes

Whether it has been a couple months or a couple days into your GMAT prep, finding innovative ways to incorporate GMAT prep into your daily routine can be vital to achieving the best score possible. Whether it is during your morning commute, or while getting ready for bed, here are 5 tips which can enhance your GMAT score in under 20 minutes

1. Read, LISTEN, repeat 

Some of the most handy ways to integrate GMAT learning into your everyday schedule is to use your eyes and ears. Pick-up a Newspaper, or start listening to a political podcast. These types of mediums are full of new vocabulary that you may encounter on the GMAT verbal section. If possible, we suggest writing down your new words in a separate notebook, including definition and usage. Keep this list on you, and review and refresh your memory when you get the chance. This type of exercise is great for quick learning, as you can expand your vocabulary while commuting into work or while going on a daily run. 

2. Practice Reviewing

So let’s assume you have listened to a podcast, or read an article. The next best thing, besides writing down any new vocabulary, is to practice rewriting what you have just heard (or read) in your own words. We suggest spending 10 minutes writing a summary of what you just heard or read. Then, review your work and make corrections where necessary. Try to put your newly learned vocabulary into practice during this exercise as well! This little trick is something you can do in under 20 minutes, and will help you put into practice your newly learned vocabulary while strengthening the part of your brain that deals with writing and sentence structure. 

3. Flashcards (for quant!?)

Yes, flashcards may seem cliché when it comes to studying for tests, but they work! In addition to using flashcards to memorize vocabulary, you can also use flashcards to memorize necessary math formulas. Write down tricky math formulas which you may find useful for the exam. While riding the train to work, or while brushing your teeth, flip through the flashcards! 

4. Get a Study Buddy 

Find someone who is also taking the GMAT exam and find time to study with them! Just meeting up for 20 minutes can help you get more comfortable with the exam and ask questions about their GMAT prep. Even if you are not actively studying! Grabbing coffee and complaining about the rigors of studying with a fellow GMAT test-taker. This can be a huge anxiety release (for both of you!). By sharing your studying experiences, you may even pick-up some new tricks yourself. 

5. Focus on Your Mental Health

Take those 20-minute breaks (whether alone or with a buddy). Meditate, breathe, take a walk. These simple breaks can help you succeed in the long run. Being anxious about the upcoming exam is normal. But filling up with stress won’t help you much on the day of the exam. This is why finding opportunities to study that don’t feel like studying can be helpful for your mental health. Listening to podcasts and reading a magazine article can be soothing. If you enjoy doing math, then simply jotting down some practice problems while waiting at a restaurant or before going to bed can all be things which help alleviate your stress while strengthening your GMAT knowledge. 

 

Regardless of where you are in your GMAT journey, we here at ApexGMAT are here to help. We offer 30-minute complimentary consultation calls with all interested GMAT studiers. You can contact us here! 

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Best Practices That Can Lead To Successful GMAT Performance
Posted on
14
Dec 2021

Best Practices That Can Lead To Successful GMAT Performance And MBA Acceptance

As the GMAT exam is a key step in being accepted to top MBA programs, maintaining some practices and utilizing those can be pretty effective. Successful GMAT performance is a determining feature in your MBA career, therefore superb assurance is relevant for a smooth preparation path. 

Each section of the GMAT test has more tiny nuances than you know, so each requires preliminary research and a meticulous approach. Hence, to succeed on the GMAT, you must familiarize yourself with the types of questions and take practice tests. In this article, we aim to introduce you to some tips that are essential for your GMAT performance and MBA acceptance. 

1. Remember what you are taking the GMAT for

Preparing for and taking the GMAT is stressful and time consuming. But you are preparing for it in order to get accepted to your desired MBA program. Remember why you are taking the GMAT. The path that you are taking right now is a long journey, which will lead you into your brightest future. Being successful means that you will be granted ample opportunities and doors for your professional prosperity will open!

2. GMAT is a marathon and not a sprint

We get it. The GMAT is a long 3 hour and 30 minute test that makes you anxious. However, those hours are building blocks to your future career. You cannot just jump over the materials without diving into each one. Every GMAT section requires meticulous thought and preparation. GMAT mostly checks your endurance and psychological tolerance. You are not supposed to know everything. However, you are supposed to behave appropriately in connecting the dots of the exam and focusing on what you see on the exam. Be confident in your selected answers but make sure to double check your responses. Even reading the question very carefully is a time-consuming task, but at least you know what is being asked rather than skimming through the questions and getting the answers incorrect for the sake of finishing the exam early. 

3. Pick your MBA program before, or while, preparing for the GMAT

Find a school and program which fits your desires, goals, and aspirations! As we mentioned above, you should know why you are taking the GMAT. This includes knowing which school(s) you are hoping to apply to. For example, the dream of studying at Harvard can urge you to work harder and put in more effort. We recommend that you have a goal in mind of where you want to see yourself in the near future. 

4. The GMAT is intense. So is business school

We do understand that GMAT preparation can be stressful and at the same time intense. But your future goals might be more challenging. Business schools and top MBA programs require you to develop high endurance. You learn valuable skills during your study prep which will serve you well during your MBA and professional career. Remember that you create your own path to MBA acceptance. This means accepting every single difficulty with high confidence.

5. Have a clear definition of your GMAT goals

A good practice for successful GMAT performance can be to consider your long-term goals and vision. You can think of this as a mission statement for yourself to consider why your goals exist. In addition to all these long-term goals, remember that GMAT falls into this category as well. The GMAT journey is an arduous one, but you undertake it in order to succeed in your goals. 

6. Develop GMAT tricks and self-cheating

You need to have a list of tricks and cheating strategies for each section. For instance, for the GMAT Quant section, you may plug in the numbers to determine the correct answer. In this case, if you are not sure about the correct answer, make some strategic assumptions which will help you work through the problem. When it comes to the Data Sufficiency section try the trick of the elimination method. For the Integrated Reasoning section, keep track of the relevant information, there is no need to know everything. Eventually, the Analytical Writing Assessment will require you to come up with a plan or an outline and spend some time on digging deeper into the material. 

7. Retake the GMAT if needed

We do realize that it might sound intimidating to take the GMAT exam a second time or even more, but if you don’t have the score needed, it is worth going through the process. Retaking the GMAT will surely increase your self-awareness. To get into some prominent MBA programs, your score needs to be in a certain range. It may not come easy, but the GMAT is necessary in this case. Being able to demonstrate your knowledge based on a high GMAT score is vital in succeeding at any university. You will be working in a business environment, hence you should be true to yourself and look for the MBA opportunity that is the perfect fit for you. It is easy to substitute between schools, but you need to be specific about one or some few schools and strive for excellence for those ones especially. MBA programs are seeking candidates that are more than “great on paper”. Resilience, persistence, demonstrated collaboration, and job experience with promotions are all important signs of program success. You can satisfy the majority of those features with a perfect GMAT score. The final result is the most important one. It does not matter how many times you have taken the exam. 

8. Make yourself the conqueror

Your inner beliefs and thoughts are more important than anything else. There is an old saying “fake it, till you make it”, and the same can apply for the GMAT exam and MBA programs. It might be super hard to pull yourself out of your comfort zone to uncover every single thing about the GMAT, but convince yourself that everything is under control. The more you panic-  either about studying or managing your time – the worse it turns out to be for your mental health. If you give yourself credit, even for the tiniest thing, you will see that things fall in place. You’ll be in a position that you once might have thought impossible to achieve. 

9. GMAT is not a hindrance, it’s a ladder!

The majority of the students tend to consider the GMAT as a bog that pulls them down to drown with stress and irritated nerves, however that is not always the case. The GMAT is the thing that determines your MBA acceptance, which therefore provides you with ubiquitous pride and chances to thrive as a person. Instead of avoiding that, immerse yourself in that whole process. The more you sink the harder you need to work to get out. A good GMAT score can bolster your place in any school. With proper preparation comes the ability to absorb more as a test taker and student. You must strike a balance in your approach and skills to succeed on the GMAT. Like climbing a ladder, it takes effort to reach the top. The GMAT is there to help you, not to hinder you. Seek it, then make it. 

10. Make the uncertain certain

The road to business school can be long and winding, and it can also be fraught with uncertainty. When you first start crafting your application piece by piece, you never know what will happen in the future. You are taking small steps towards your major goal. What if you actually could make it a reality? It is not that hard, the only thing is introducing yourself as a go-getter with gaining experiences that will undoubtedly lure the admissions officer. 

First of all, as we mentioned earlier, know the “reason”. This is where an MBA application differs from most other graduate programs. You must not only demonstrate that you are academically and professionally prepared, but you must also clearly express your long-term career goals and how an MBA will assist you in achieving them. In order to make everything certain, you need to create yourself, sometimes from scratch. Develop the “how” strategy. Design yourself, create and then become.  

 

Final Thoughts

We are sure you are already familiar with the GMAT exam and maybe you are now preparing. However, it is always necessary to come up with some best practices that can lead to successful GMAT performance which then results in quiet and peaceful MBA acceptance without any hurdles. In this article, we tried to gather some practices and tricks that you can make use of for your overall preparation process and success. One day you will become the achiever of your dreams and acquire the best in this world. The significant and essential thing is believing in yourself and walking into every situation proudly and positively.

 

Contributor: Ruzanna Mirzoyan

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How-to GMAT: No Calculator? Use These Mental Math Tips Instead
Posted on
07
Sep 2021

How-to GMAT: No Calculator? Use These Mental Math Tips Instead

The GMAT is an exam largely focused on numbers and numerical data. And while doing math on the GMAT should be avoided sometimes it is inevitable. True, the test-taker is given a calculator for the duration of the Integrated Reasoning section but the same cannot be said for the Quantitative Reasoning Section. 

This article is going to provide some smart calculation shortcuts and mental math tips to help you go through some arithmetical questions without losing too much time and help you get a higher score on the GMAT Quant.

The Basics

Before explaining any methods for dividing and multiplying with ease, let’s make sure we have revised a few simple rules:

  • Numbers with an even last digit are divisible by 2 – 576 is and 943 is not;
  • Numbers with a sum of digits divisible by 3 are also divisible by 3 – 3,465 for example (3+4+6+5=18);
  • If the last 2 digits of a number a divisible by 4, the number itself is divisible by 4 – 5,624 for example (because 24/4=6);
  • Numbers with last digit 0 or 5 are divisible by 5;
  • Numbers that can be divided by both 2 and 3 can be divided by 6;
  • Similar to numbers divisible by 3, numbers divisible by 9 must have a sum of digits divisible by 9 – 6,453 for example;
  • If the last digit of a number is 0 it is divisible by 10;

With that out of the way, we can move onto some more advanced mental math techniques.

Avoid division at all costs

Don’t divide unless there is no other option. And that is especially true with long division. The reason why long division is so perilous is that it is very easy to make a careless mistake as there are usually several steps included in the calculation, it takes too much time, and to be honest, few people are comfortable doing it.

Fortunately, the GMAT doesn’t test the candidates’ human-calculator skills but rather their capacity to think outside the box and show creativity in their solution paths, especially when under pressure – exactly what business schools look for.

However, sometimes you cannot avoid division, and when that is the case remember: Factoring is your best friend. Always simplify fractions especially if you’ll need to turn them into decimals. For example, if you have 234/26 don’t start immediately trying to calculate the result. Instead, factor them little by little until you receive something like 18/2 which is a lot easier to calculate.

A tip for factoring is to always start with smaller numbers as they are easier to use (2 is easier to use compared to 4, 6, or 8) and also look for nearby round numbers. 

If you have to calculate 256/4 it would be far less tedious and time-consuming to represent 256 as 240+16 and calculate 240/4+16/4=60+4=64. Another example is 441/3. If we express it like 450-9 it is far easier to calculate 450/3-9/3=150-3=147.

Dividing and Multiplying by 5

Sometimes when you have to divide and multiply by 5 (you’ll have to do it a lot) it would be easier to substitute the number with 10/2. It might not always be suitable for your situation but more often than not it can be utilized in order to save some time.

Using 9s

With most problems, you could safely substitute 9 with 10-1. For example, if you have to calculate 46(9) you can express it as 46(10 – 1) which is a lot more straightforward to compute as 46(10) – 46(1) = 460 – 46 = 414

You can also use the same method for other numbers such as 11, 8, 15, 100, etc:

18(11) = 18(10 + 1) = 180 + 18 = 198

28(8) = 28(10 – 2) = 280 – 56 = 224

22(15) = 22(10 + 5) = 220 + 110 = 330

26(99) = 26(100 – 1) = 2600 – 26 = 2574

Dividing by 7

The easiest way to check if a number is divisible by 7 is to find the nearest number you know is divisible by 7. For instance, if you want to check if you can divide 98 by 7 you should look for the nearest multiple of 7. In this instance either 70, 77, or 84. Start adding 7 until you reach the number: 70 + 7 = 77 + 7 = 84 + 7 = 91 + 7 = 98. The answer is yes, 98 is divisible by 7 and it equals 14

Squaring

When you have to find the square of a double-digit number it might be easier to break the number into components. For example, 22^2 would be calculated like this:

22^2
= (20 + 2)(20 + 2)
= 400 + 40 + 40 + 4
= 484

Similarly, if you have to find the square of 39 instead of calculating (30 + 9)(30 +9) you could express it like this:

39^2
= (40 – 1)(40 – 1)
= 1600 – 40 – 40 + 1
= 1521

You can use the same approach when multiplying almost any double-digit numbers, not only squaring. For example 37 times 73:

(40 – 3)(70 + 3)
= 2800 + 120 – 210 – 9
= 2701

Conclusion

This ends the list of mental math tips and tricks you can utilize to make the Quant section a bit less laborious. Keep in mind that no strategy or shortcut would be able to compensate for the lack of proper prep so it all comes down not only to practicing but doing so the right way.

For more information regarding the GMAT Calculator, GMAT Calculator & Mental Math – All You Need To Know, is a very insightful article to read.

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