Question

If x and y are odd integers,then prove that x^2+y^2 is even but not divisible by 4.

Answer 1

hello there !!

we know that every odd positive integer is of the form 2q+1 for some integer q
so, let x=2p+1 and y=2r+1 for some integers p and r .

=> x²+y²= (2p+1)²+(2r+1)²

=> x²+y²= 4(p²+r²)+4(p+r)+2

=> x²+y²= 4[(p²+r²)+(p+r)]+2

=> x²+y²= 4a+2 where a = [(p²+r²)+(p+r)]

x²+y² is even and leaves remainder 2 when divided by 4