The Perfect Square
You must have heard that the circle is the perfect shape but do you know about perfect squares?
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.
. 1764
2 882
2 441
3 147
3 49
7 7
7 1
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.
Remember:
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?
21
35
54
105
150
11 comments
Tushar on October 25th, 2012 at 2:01 am
105?
Shameilia on October 25th, 2012 at 10:21 am
I got 105 as well. 4 3's, 2 5's, and 2 7's
Rahul Sehgal on October 26th, 2012 at 11:49 pm
Could somebody please share the answer of this one ? I am also getting 105 as the answer for this one...
This is what i did

945s = 945 x S = 3^3 x 5 x 7 x S. So, basically, we want to have a number that can result in even powers of 3, 5 and 7 as these are currently having odd powers in 945 and moreover there can be anny additional factors of 'S', but those also have to be 'even' in number also. The only option that does that is '105' which can be factorized further as  5 x 7 x 3. This then combined with 945, makes the expression as  3^4 x 5^2 x 7^2 with all the numbers in even powers and that is waht makes it as a perfect square.
Please do let me know in case I am wrong in my approach...
Arun on October 29th, 2012 at 11:49 am
Trick used is not clear. Confusing
Shane on October 31st, 2012 at 7:58 am
Its easy. 945 breaks out into primes of 3,3,3,5,7. You need an even number of primes for 3,5,and 7. so add one 3, one 5, and one 7. Multiply them you have 105
Ryan on November 2nd, 2012 at 11:18 am
Wow awesome I got 105. This really clarifies the perfect square concept one of those things that I usually just do in my head. It's so much better to have a method for it will definitely save me time.
Thanks!
Bazlur Rashid on January 22nd, 2014 at 8:33 am
How do I make a natural number to the sum of two or three
perfect square.
kay on April 10th, 2014 at 11:49 pm
get each number's total number of factors and then add them to 945's total number of factors (6), to be perfect squares all factors must be an odd number only answer is 105/6+3 = 9
umashankar on May 4th, 2014 at 2:56 am
but 64 is a perfect square and can be written as 4*4*4 which is odd number of equal "integers"...so the abovesaid rule goes wrong here, right?i think it must be said that ,there should be even number of equal prime factors in the number for it to be a perfect square.......correct me if i am wrong...
honey on May 8th, 2014 at 10:21 am
No this is not correct.. see manually 105 is not matched with 945..
vipul bhasin on July 16th, 2016 at 6:16 pm
What if we have some larger number say, Is 84791764 a perfect square?
I guess, prime factorization will not be feasible in this case as it will consume lots of time.
Is there any other way to find the answer to this question?