Question

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

Answer 1

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Let the two odd numbers be
(2a+1) & (2b+1) because if we add 1 to any even no. it will be odd.
x²+y²
=>(2a+1)²+(2b+1)²
=>(4a²+4a+1)+(4b²+4b+1)
=>4(a²+b²+a+b)+2
4 Is not a multiple of 2 it means clearly that 4 is not multiple of x²+y² , so x²+y² is even but not divisible by 4.
Hence proved.
@:-)

Answer 2

Since x and y are odd positive integers so

Let x = 2n + 1 and y = 2m + 1

x² + y² = (2n + 1)² + (2m + 1)²

= 4(n² + m²) + 4(n + m) + 2

= 4 {(n² + m² + n + m}) + 2

= 4q + 2

Where q = n² + m² + n + m is an integer

Since

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

Not divisible by 4