Question

If m & n are odd positive integer then m²+n² is even but not divisible by 4, justify

Answer 1

Explanation:

Let m = 2a+1, and n=2b+1. Therefore,
`m^2+n^2=4a^2+4a+1+4b^2+4b+1=4(a^2+b^2+a+b)+2`. Therefore, this is always an even number, because you can factor a 2 out. However, it is not a multiple of 4 because of the 2 at the end; this result is always 2 modulo 4.