Question

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

Answer 1

let m = 2 x + 1  
and  n = 2 y + 1   
(where x and y are non-negative integers.)

m² + n² = (2x+1)² + (2y+1)² 
m² + n² = 4 x² + 4x + 1 + 4 y² + 4y + 1 
m² + n² = 4 (x² + y² + x + y ) + 2 

now divide m² + n²  by 4, then we get a reminder 2 and quotient (x² + y² + x + y ).
 
So m² + n² is divisible by 4