We have given two integers x and y and it is also given that both x and y are positive odd integers and we have been asked to prove that x²+y² is even number but it is not divisible by 4.

Kindly give me a detailed solution of this question

RS Aggarwal, Class 10, chapter 1A, question no 9

## We know that any odd positive integer is of the form 2q+1, where q is an integer.

So, let x=2m+1 and y=2n+1, for some integers m and n.

we have x²+y²

x²+y²=(2m+1)²+(2n+1)²

x²+y²=4m²+1+4m+4n²+1+4n=4m²+4n²+4m+4n+2

x²+y²=4(m²+n²)+4(m+n)+2=4{(m²+n²)+(m+n)}+2

x²+y²=4q+2, when q=(m²+n²)+(m+n)

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

x²+y² is even but not divisible by 4.