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

Let m = 2 x + 1 and n = 2 y + 1 where x and y are non-negative integers or positive integers

Now,

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

So, on dividing m² + n² by 4, 2 will be the remainder and a quotient would be (x² + y² + x + y ).

So m² + n² is even but not divisible by 4