Question

Prove that if x and y are odd positive integers then x2+y2 is even but not divisible by 4.

Answer 1

Step-by-step explanation:

x and y are odd positive integers:

then x=2a+1 and y=2b+1

x2+y2=(2a+1)^2+(2b+1)^2=4a^2+4a+1+4b^2+4b+1=

4(a^2+b^2+a+b)+2= 4k+2

So the resultant number is even but when divided by 4 leaves reminder of 2

Answer 2

let us take any x and y as any positive odd integers

so,

x=2m+1

y=2n+1

X2 +Y2=(2m+1)2 +(2n+1)2

4m2 + 1 + 4m + 4n2 + 1 + 4n

4(m2 + m + n2+ n) + 2

4q+2

(where q= m2 + m + n2 + n)

hence, we can say that X2+Y2 is even but not divisible by 4