Welcome back to our series on number properties. In the last article, we learned how to find the greatest common factor/divisor of a set of integers. This article will cover the related topic of the least common multiple.
Least common multiple
The least common multiple (LCM) of a set of integers is the lowest integer that is divisible by each integer in the set. The LCM is the product of all the prime factors of each integer in the set – without repetitions between members of the set.
Here’s what that qualification means: if one integer has three prime factors of 2 and another integer has four prime factors of 2, the LCM of these integers has four – not seven – prime factors of 2. Don’t add up the numbers of prime factors of 2 – just take the highest one you can find. The LCM has only as many “copies” of each prime factor as can be found in any single member of the set.
Let’s use an official GMAT quant problem to practice finding an LCM:
If M is the least common multiple of 96, 196, and 300, which of the following is NOT a factor of M?
(A) 600
(B) 700
(C) 900
(D) 2,100
(E) 4,900
To start, let’s prime factorize 96, 196, and 300.
96 = 25 * 3
196 = 22 * 72
300 = 22 * 3 * 52
To list the prime factors of the LCM, find the highest exponent attached to each prime factor. That is, the highest number of prime factors of 2 belonging to any member of the set, the highest number of prime factors of 3 belonging to any member of the set, so on and so forth.
LCM = 25 * 3 * 52 * 72
For this problem, we don’t need to perform the multiplication to find the actual value of the LCM – the list of prime factors is more useful! We were asked which answer choice is NOT a factor of M, this LCM.
To figure this out, we can apply “factor logic”. The answer choice that is not a factor of our LCM is the one that has a factor not shared by the LCM. If any answer choice has more than five factors of 2, more than one factor of 3, more than two factors of 5, or more than two factors of 7, it can’t be a factor of the LCM, meaning it’s the correct answer choice. Let’s review the options:
(A) 600
(B) 700
(C) 900
(D) 2,100
(E) 4,900
Can you spot the culprit? If you have to create a prime factor tree on scratch paper for each of these answer choices in order to solve this problem, you probably need more practice with number properties problems. These numbers should be easy enough to “break down” to prime factors mentally.
600 = 22 * 52 * 2 * 3
700 = 22 * 52 * 7
900 = 22 * 52 * 32
2,100 = 22 * 52 * 3 * 7
4,900 = 22 * 52 * 72
A number ending in double zeros indicates two factors of 10, or two prime factors of 2 and two prime factors of 5. So these numbers are easy to work with. The correct answer choice is C, because 900 has two prime factors of 3, while the LCM we found has only one prime factor of 3.
Here’s a different kind of official LCM problem:
M and N are integers such that 6 < M < N. What is the value of N?
- The greatest common divisor of M and N is 6.
- The least common multiple of M and N is 36.
(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not sufficient.
This problem involves both GCD and LCM elements. Statement 1 is clearly insufficient by itself, because there is no “upper limit” for M and N. There are many possibilities for their values.
Statement 2 stands a better chance of being independently sufficient. The LCM of M and N is 36, and we know from the question stem that both integers are greater than 6. We need to consider the fact that an LCM can be equal to an integer in the set. Every number is both a factor and a multiple of itself (using multiplication by 1), so this is allowed. Since this is true, N could be 36, while M could be any other factor of 36 like 9, 12, or 18. (The factors 1, 2, 3, 4, and 6 are ruled out by the constraint in the question stem that 6 < M < N.) If 36 is the only possible value for N, then statement 2 is sufficient. But 36 is not the only possible value for N. N could also be 18 if M is 12, or 12 if M is 9.
So statement 2 is insufficient, and we need to consider both statements together. The best step at this point is to list all the possibilities for M and N that were allowed by statement 2:
N = 36 and M = 9, 12, or 18
N = 18 and M = 12
N = 12 and M = 9
If only one of these possibilities is also allowed by statement 1, then the statements together are sufficient (answer choice C). For how many of these possibilities is 6 the GCD of M and N? Nothing from the N = 36 row will work, because in any of these cases, the GCD is equal to M, which is not 6. N = 18 and M = 12 yields a GCD of 6, while N = 12 and M = 9 yields a GCD of 3. So the only value of N allowed by both statements is 18, meaning that they are sufficient together. The correct answer is C.
Let’s look at one more official LCM problem:
If a and b are positive integers, what is the value of the product ab?
- The least common multiple of a and b is 48.
- The greatest common factor of a and b is 4.
(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not sufficient.
This time we are asked for the product of the two integers rather than the value of the larger one. But we can take the same approach as before. See which scenarios are allowed by the LCM statement (this time it is statement 1 instead of statement 2) and then see if more than one of these scenarios is also allowed by the GCF statement.
First, we need all the pairs of integers whose LCM is 48.
48 and any factor of 48
16 and 3, 6, 12, or 24
Is there more than one possibility from this list that has a GCF of 4? Looking at the 16 row, the only pair with a GCF of 4 is 16 and 12. But there is also a possibility from the 48 row! 48 and 4 also have a GCF of 4. It would be understandable to select answer choice E at this point, but it would be a mistake. Remember that this problem asked us about the product ab, not about the individual values of a and b. We will have to see whether 48 * 4 and 16 * 12 have the same product. As it turns out, they do: 192.
This was a fun and helpful exercise, but there is a little-known rule about a set’s LCM and GCF that could have made this problem much easier: The product of any two positive integers is equal to the product of their LCM and their GCF.
a * b = (LCM of a and b) * (GCF of a and b)
Why is this true? Let’s look at an example with the integers 12 and 18. This will serve as a recap of what LCM and GCF are.
12 = 22 * 3
18 = 2 * 32
12 * 18 = 216
The product of the LCM and GCF of 12 and 18 should equal 216. Let’s check:
LCM = 22 * 32 = 4 * 9 = 36
GCF = 2 * 3 = 6
LCM * GCF = 36 * 6 = 216
It worked! But why? Well, take a look at the prime factors that were selected to derive the LCM and the GCF. To find the LCM, we took the highest number of prime factors of 2, the 22 from 12, ignoring the single prime factor of 2 in 18. We also took the highest number of prime factors of 3, the 32 from 18, ignoring the single prime factor of 3 in 12. Then when we took the GCF, we used the integers that we ignored in our selection for the LCM.
When there are only two integers in the set, any common prime factors left out of the LCM group will be taken for the GCF, and any unshared prime factors left out of the GCF group will be taken for the LCM. Therefore the product LCM * GCF represents the multiplication of the full prime factorizations for each integer. Since these prime factors are the “building blocks” of each integer, the product of all these primes is equal to the product of the two integers.
Try your own examples in order to solidify your understanding of this property. This concludes our study of least common multiple and greatest common factor. In the next article, we’ll move onto a different kind of number properties topic: place value.
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Contributor: Elijah Mize (Apex GMAT Instructor)