Welcome back to our series on number properties. Now that you know about prime factors, it’s time to learn a special property about the prime factors of perfect squares. Perfect squares are integers that represent the product resulting when some integer is multiplied by itself. This “multiplication by self” is called “squaring” and can be notated with an exponent of 2.
52 = 25
112 = 121
172 = 289
The integer that is squared to produce the perfect square is called that number’s square root. 5 is the square root of 25. 11 is the square root of 121. 17 is the square root of 289.
Numbers that are not perfect squares have square roots too, but their square roots are not integers. The square root of 2 is about 1.41. Since 1.41 is not an integer, 2 is not a perfect square. The square root of 3 is about 1.71. Since 1.71 is not an integer, 3 is not a perfect square.
The only numbers that don’t have square roots are negative numbers, since no value multiplied by itself produces a negative number. A positive value squared produces a positive, and a negative value squared produces a positive. 02 = 0.
Prime factors of perfect squares
In the multiplication table below, the first 20 perfect squares form the diagonal line highlighted in yellow:
For GMAT quant, it’s helpful to know this series of numbers. Perfect squares are an important topic on GMAT quant, and one frequently-tested property is that perfect squares have prime factors in pairs. Another way to say this is that if a perfect square has a certain prime factor, it has an even number of that prime factor.
In the prime factorization of the perfect square, all of the exponents used to notate the number of occurrences of each prime factor are even. A perfect square cannot have five prime factors of 2. It must have two or four or six (etc.) prime factors of 2. A perfect square cannot have three prime factors of 5. It must have two or four or six (etc.) prime factors of 5.
This makes logical sense. If an integer has an odd number – or an “uneven number” – of some prime factor, it will be impossible to evenly divide the occurrences of this prime factor into two groups. Therefore the integer in question cannot be the product of two equal integers – also known as the square of an integer.
Let’s explore this property with an official GMAT problem:
If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be
(A) 2
(B) 5
(C) 6
(D) 7
(E) 14
In order to make a perfect square, we need even numbers of each prime factor. Therefore if 3,150 has odd numbers of one or more prime factors, we need y to contain the “missing” primes such that 3,150 * y has even numbers of these prime factors. If this doesn’t make sense, observing the prime factorization of 3,150 may help.
3,150 = 2 * 32 * 52 * 7
3,150 has a pair of 3s and a pair of 5s, but only one prime factor of 2 and one prime factor of 7. This means that there is no way to split the prime factors of 3,150 into two identical groups. Therefore 3,150 is not the product of two identical integers, i.e., not the square of an integer.
What prime factors are “missing” in order to make this possible? We need at least one more prime factor of 2 and one more prime factor of 7. Therefore, in order for 3150 * y to be the square of an integer, y must have at least one prime factor of 2 and one prime factor of 7. The only answer choice that satisfies this condition is (E) 14. No lower number can work.
Another property of the factors of perfect squares is tested in the following official problem:
If 2x+y = 48, what is the value of y?
- x2 = 81
- x – y = 2
(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.
Some knowledge of exponents helps make sense of this problem. We can apply a “change of base” to 48:
2x+y = 48
2x+y = (22)8
2x+y = 216
x + y = 16
This simplifies the equation we are given in the question stem. It is not absolutely necessary to perform these steps in order to answer the problem correctly – you could simply count the variables and count the equations. Since there are two variables, x and y, a system of two different equations relating to these variables will be sufficient to solve both variables. The question stem supplies one such equation, and statement 2 supplies another. Therefore statement 2 by itself is sufficient. But what about statement 1 by itself?
If you said that statement 1 by itself is sufficient because it supplies the value of x (x = 9) such that the value of y can also be determined using the equation in the question stem, you probably forgot about a property of the factors of perfect squares. If x2 = 81, then yes, x could equal 9, the positive square root of 81. But x could also equal -9, the negative square root of 81. 9 * 9 = 81, and -9 * -9 = 81. Unless we are told in some way that x is positive – and in this problem, we aren’t told this – either value is possible. If there are two possible values for x, then there are two possible values for y, and statement 1 by itself is insufficient.
Note that x2 doesn’t have to be a perfect square for this property to apply. Every positive number has both a positive and a negative square root. 0 has only one square root: 0.
Here’s a final official GMAT problem to check your comprehension of this “negative square root” property.
If k2 = m2, which of the following must be true?
(A) k = m
(B) k = –m
(C) k = |m|
(D) k = -|m|
(E) |k| = |m|
The correct answer is E. We don’t know about the positivity or negativity of either of the square roots k and m. But we know that if the squares of k and m are equal, k and m must be equal in absolute value.
This concludes our discussion of the factors of perfect squares. In the next article, we will look at the special case of consecutive perfect squares.
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