Question

if m and n are odd positive integers, then m²+n² is even, but not divisible by 4. Justify.

Answer 1

Let us take:
m² = 3 and n² = 5.
So, 3² = 9
       5² = 25.
Therefore, 3²+5² = 9+25 =34 which is even.
Hence, we verified that 34 is a even no. but is not divisible by 4.

Answer 2

An odd number can be write as:
2k+1, where k is integer
If k ≥ 0, integers 2k+1 are positive.
m=2k+1
n =2p+1
m² + n² = (2k+1)² + (2p+1)² = 4k²+4k+1 + 4p²+4p+1 = 4(k²+k+p²+p) + 2 = 4q + 2
4q + 2 = 2(2q + 1) this number is divisible by 2, but not by 4, because 2q+1 is odd for all q.
For negative odd integers this theorem is true too, because for all numbers (-m)² = m².