Question

Question 7
Prove that, if x and y are both odd positive integers then $x^2+y^2$ is even but not divisible by 4.

Answer 1

Let x = 2m + 1 and y = 2k + 3 are odd positive integers, for some positive integer m, k.
Then, $x^2+y^2=(2m+1{)}^2+(2k+3{)}^2$
= $4m^2+1+4m+4k^2+9+12k[\because (a+b{)}^2=a^2+2ab+b^2]$
$=4(m^2+k^2)+4(m+3k)+10=even$
$=4\left[\right(m^2+k^2)+(m+3k)+2]+2=even$
$\Rightarrow$ On dividing $x^2+y^2$ by 4 , it leaves 2 as the remainder [division algorithm]
$\Rightarrow$ $x^2+y^2$ is not divisible by 4 .
Hence, $x^2+y^2$ is even for every odd positive integer x and y but not divisible by 4.