Which of the following integers is the square of an integer?

Which of the following integers is the square of an integer?

(A) 73,410,624
(B) 63,398,748
(C) 54,113,892
(D) 42,785,337
(E) 31,298,793


Solution - Square the each number from 1 to 9 and check the units digit for match. Only 2^2 and 8^2 result to unit digit 4. Hence answer A.

Thanks,

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Simply check the square of each integer from 1 to 100
1=1
2=4
3=9
4=16
5=25
6=36
7=49
8=64
9=81
10=100

Therefore, unit's digit can't be 8,2,7 or3
Answer A

Perfect Square only ends with 0,1,4,5,6,9
Using above clear answer A

MANHATTAN GMAT OFFICIAL SOLUTION:

One way to approach this problem is to factor all of the answer choices. The one with matching pairs of prime factors is a perfect square. Although valid, this approach would be very time-consuming.

There is a backdoor solution: examine the units digit of the answers. Since the units digit of x^2 is simply the units digit of the square of the units digit of x, a perfect square must have the same units digit as one of 0, 1, 4, 9, 16, 25, 36, 49, 64, or 81. That is, a perfect square must have a units digit of 0, 1, 4, 5, 6, or 9. A perfect square cannot have a units digit of 2, 3, 7, or 8. Thus, answer choices B, C, D, and E are disqualified.

By the way, 73,410,624 is (8,568)^2. You would never be expected to figure that out on the GMAT.

The correct answer is A.

Let’s determine the possible units digits of all perfect squares:

0^2 = 0

1^2 = 1

2^2 = 4

3^2 = 9

4^2 = 6

5^2 = 5

6^2 = 6

7^2 = 9

8^2 = 4

9^2 = 1

So perfect squares can have units digits of 0, 1, 4, 5, 6, and 9. Thus, the only possible perfect square is choice A.

Answer: A

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