Question

prove that if x and y are odd positive integers then X square + Y square is even but not divisible by 4​

Answer 1

Let the two odd positive numbers be

x = 2k + 1 a nd y = 2p + 1

Hence

x²+ y² = (2k + 1)² + (2p + 1)² = 4k² + 4k + 1 + 4p² + 4p + 1 = 4k²+ 4p² + 4k + 4p + 2 = 4(k²+ p² + k + p) + 2

Clearly notice that the sum of square is even the number is not divisible by 4

Hence if x and y are odd positive integers, then x² + y² is even but not divisible by 4

Answer 2

Step-by-step explanation:

we know that any odd positive integer is of the form 2q+1 for some integer q.

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

•°• x²+y²= ( 2m+1)²+(2n+1)²

==> x²+y²=4(m²+n²)+4(m+n)+2

==> x²+y²= 4{(m²+n²)+(m+n)} +2

==> x²+y² = 4q +2 , where 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.

HENCE PROVED✔✔