Which of the following is a perfect square? GMAT Problem Solving

Question: Which of the following is a perfect square?

(A) 97,474

(B) 1,23,301

(C) 1,37,641

(D) 1,70,567

(E) 1,77,243

‘Which of the following is a perfect square?’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide 2022”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

There is only one approach to solve the problem.

GMAT usually does not want the candidates to calculate the square roots. It would take too much time to calculate the square roots of all the options.

So the candidates should try to eliminate wrong choices by some trick.

We have listed a few of them, which are:

1) Units digit: The units digit of a perfect square will always be 0, 1, 4, 5, 6, or 9. Any number having a units digit of 2, 3, 7 or 8 will not be a perfect square.

2) Even number: If there is an even number, last two digits should be divisible by 4.

3) Multiple of 3: The sum of digits if divisible by 3 and not by 9 will not be a perfect square.

D and E cannot be perfect square as last digits are 7 and 3.

A cannot be a perfect square as last two digits 74 is not divisible by 4.

However, the candidates need to think of some technique to eliminate one of B or C. It would require to be calculated.

1) Now B is not divisible by 2, 3, 5, or 11 by observation itself. so let us try with the prime 7, 13 and then 17. It is divisible by 17.

=>123,301
= 17*7253.
But 7253 is not divisible by 17.

2) Another method could be approximation.
90000<123301<160000, almost in middle of two perfect squares.
So, it should be between 300^2 and 400^2, and closer to 350 and having 1 or 9 as units digit.

Let us try
351
=> 351*351
=123,201.

So, 123,301 is not a perfect square.
Hence, C is the correct answer.

Correct Answer: A

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