The Perfect Square
Many GMAT questions employ the use of the concept of the perfect squares. So let’s discuss, what exactly is a perfect square.
A perfect square is an integer which is itself the square of another integer. In other words, a perfect square is an integer with an integer square root.
Common perfect squares are: 4, 9, 16, 25, 36, and so on
It is useful to memorize the squares of the first 20 integers as these can save you the trouble of time consuming calculations. However, you cannot memorize all perfect squares and it is important to learn how to spot perfect squares.
Let’s try an example:
Is 1764 a perfect square?
Before we begin to answer the above question, let’s ask ourselves: how do we know if 4 is a perfect square?
You can say we know 4 is a perfect square because . Likewise, 9 is a perfect square because , and 16 is a perfect square because .
Notice something yet?
Perfect squares are products of two integers that are equal. But do perfect squares always have to be products of two equal factors? While, 16 is equal to , 16 also equals . In this case, a perfect square is the product of 4 2’s. What about 64? True but you can also express 64 as . So, now you have a perfect square that is the product of 6 equal integers i.e. 6 2’s.
Do you see a pattern here?
Perfect squares are products of an even number of equal integers.
How about we apply this to 100? so 100 is a perfect square because it is the product of 2 (an even number) equal integers but 100 can also be written as . So, a perfect square can be the product of integers that are not equal. However, the number of equal integer factors of a perfect square must be even.
In fact, this is the key criterion for a perfect square. Any number that has an even number of equal integer factors is a perfect square.
Now, let’s apply this to 1764. Start by finding the prime factors of 1764.
Do you have an answer?
Yes, 1764 is a perfect square because it is the product of 2 2’s, 2 3’s and 2 7’s. 1764 is a perfect square because it has an even number of equal factors.
In order for a number to be a perfect square it must have an even number of equal factors. An easy way to remember this is that all factors of a perfect square must be in pairs.
Let’s see if you can solve the following problem using your knowledge of perfect squares.
If 945s is a perfect square then s can be which of the following numbers?