Question

Prove that ifx and y are odd positive integers, then x2+y2 is even but not visible by 4.

Answer 1

Suppose, the two odd numbers are x=2n-1 and y=2n+1.

Now, according to the question:

(2n-1)^2+(2n+1)^2

Expand it using the basic algebric formula i.e.

(a+b)^2=a^2+b^2+2ab and (a-b)^2=a^2+b^2-2ab

We get,

(2n-1)^2+(2n+1)^2=4n^2+4n+1+4n^2-4n+1

8n^2+2=2(4n^2+1)

Therefore, the above expression is divided by 2 means x^2+y^2 is even but not divisible by 4.