# 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?

1. 0
2. 1
3. 2
4. 3
5. 4

When this problem was given in the seniors division during the 2nd 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

1. 96 x 1
2. 48 x 2
3. 32 x 3
4. 24 x 4
5. 16 x  6
6. 12 x 8

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

1. 48 – 2 = 64
2. 24 – 4 = 20
3. 16 – 6 = 10
4. 12 – 8 = 4

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

1. [(48 + 2)/2]^2 – [(48 – 2)/2]^2  = 25^2 – 23^2 = 625 – 529 = 96
2. [(24 + 4)/2]^2 – [(24 – 4)/2]^2 = 14^2 – 10^2 = 196 – 100 = 96
3. [(16 + 6)/2]^2 – [(16 – 6 )/2]^2 = 11^2 – 5^2 = 121 – 25 = 96
4. ((12 + 8)/2]^2 – [(12 – 8)/2]^2 = 10^2 – 2^2 = 100 – 4 = 96