Question

Prove that x and y are both add positive integer then X2+Y2 is even but not divisible by 4

Answer 1

Any odd positive integer is of the form 2q+1.
Let x=2m+1 and y=2n+1 for some integer m and n.
x2 + y2 = (2m+1)2 + (2n+1)2
= 4m2 + 4m +1 + 4n2 + 4n +1
=4m2 + 4n2 + 4m+4n + 2
=4(m2 + n2 ) + 4(m+n) + 2
= 4{(m2 + n2)+(m+n)} +2
Let (m2 + n2 )+(m+n)=q
= 4q+2
So,x2 + n2 is even and leaves remainder 2 when divided by 4.
Therefore,it is even but not divisible by 4

Answer 2

Step-by-step explanation:

Since x and y are odd positive integers, so let x = 2q + 1 and y = 2p + 1 ,

•°• x² + y² = ( 2q + 1 )² + ( 2p + 1 )² .

= 4( q² + p² ) + 4( q + p ) + 2 .

= 4{( q² + p² + q + p )} + 2 .

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

•°• x² + y² is even and leaves remainder 2, when divided by 4 that is not divisible by 4.

Hence, it is solved

THANKS

#BeBrainly.