Question

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

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

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

Answer 2

Step-by-step explanation:

Since m and n are odd positive integers, so let m = 2q + 1 and n = 2p + 1 ,

•°• m² + n² = ( 2q + 1 )² + ( 2p + 1 )² .

= 4( q² + p² ) + 4( q + p ) + 2 .

= 4{( q² + p² + q + p )} + 2 .

= 4y + 2 , where y = q² + p² + q + p is an integer .

•°• q² + p² is even and leaves remainder 2, when divided by 4 that is not divisible by 4.

Hence, it is solved

THANKS

#BeBrainly.