Which number cannot be the difference between two consecutive perfect squares?

Question

A. 17

B. 29

C. 41

D. 44

Answer 1

Answer 2

When we consider two consecutive numbers, one will be odd and the other will be even.

An odd number has a square that is odd. An even number has a square that is even. Due to this property, the squares of the two consecutive odd and even numbers will also be odd and even.

Therefore, we are considering the difference between an odd number and an even number. An odd number minus and even number is always odd (for example, 3-2=1 is odd). Our question is asking which number cannot be the difference between two consecutive perfect squares, so the answer must be an even number, which is answer choice D.

Alternative solution: We see that the difference of two consecutive perfect squares can be expressed as (a+1)² - a². If we expand this expression, we get 2a+1, where a is an integer. Only an odd number can be expressed in this way. Therefore, the number that cannot be expressed in this way is even, and the only answer choice that satisfies this property is answer choice D.