Question

prove that if x and y are both off positive integer then x²+y² is even but not divisible by 4​

Answer 1

Question:

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

Solution:

Since x and y are odd positive integers, they leave remainder 1 on division by 2. So let x = 2p + 1 and y = 2q + 1. Then,

x² + y² = (2p + 1)² + (2q + 1)²

x² + y² = 4p² + 4p + 1 + 4q² + 4q + 1

x² + y² = 4p² + 4q² + 4p + 4q + 2

x² + y² = 4(p² + q² + p + q) + 2

Let p² + q² + p + q = m. Then,

x² + y² = 4m + 2 → (1)

This implies that the sum x² + y² always leave remainder 2 on division by 4. So it can never be divisible by 4.

From (1),

x² + y² = 2(2m + 1)

This means x² + y² is always an even number.

Thus the sum is always an even number but can't be divisible by 4.

Hence Proved!

#answerwithquality

#BAL