Question

Prove that if x and y are both odd positive integer then x ²+y² is even but not divisible by 4

Answer 1

Answer.

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✔✔