If k is an integer, what is the smallest possible value of k such that 1040kis the square of an integer?

BTGmoderatorLU wrote: ↑

Wed Apr 21, 2021 9:54 am

Source: Magoosh

If k is an integer, what is the smallest possible value of k such that 1040k is the square of an integer?

A. 2
B. 5
C. 10
D. 15
E. 65

The OA is E

Solution:

Let’s prime factorize 1040 first:

1040 = 10 x 104 = 2 x 5 x 4 x 26 = 2 x 5 x 2 x 2 x 2 x 13 = 2^4 x 5 x 13

In order for a number to be a perfect square, the exponent of any particular prime factor of the number must be even. Therefore, in order for 1040k to be a perfect square, it must have an even number of 5s and 13s (notice it already has an even number of 2s). Since we want k to be the smallest possible positive integer, k must be 5 x 13 = 65 so that 1040k will have two 5s and two 13s.

Answer: E