# Expressing a number as a difference of two square integers

In how many ways can 96 be expressed as the difference of two square integers?

- 0
- 1
- 2
- 3
- 4

When this problem was given in the seniors division during the 2^{nd} international Vedic Mathematics Olympiad (IVMO 2022) last November 2022, only a few managed to get the right answer.

But only a few seconds are needed to determine the answer: just list down the factor pairs of 96 and count how many of them have an even number difference.

96 can be factored as

- 96 x 1
- 48 x 2
- 32 x 3
- 24 x 4
- 16 x 6
- 12 x 8

Of those 6 pairs, 4 will have an even number difference, namely:

- 48 – 2 = 64
- 24 – 4 = 20
- 16 – 6 = 10
- 12 – 8 = 4

Therefore, there are 4 ways 96 can be expressed as a difference between two integer squares. They are:

- [(48 + 2)/2]^2 – [(48 – 2)/2]^2 = 25^2 – 23^2 = 625 – 529 = 96
- [(24 + 4)/2]^2 – [(24 – 4)/2]^2 = 14^2 – 10^2 = 196 – 100 = 96
- [(16 + 6)/2]^2 – [(16 – 6 )/2]^2 = 11^2 – 5^2 = 121 – 25 = 96
- ((12 + 8)/2]^2 – [(12 – 8)/2]^2 = 10^2 – 2^2 = 100 – 4 = 96