Question

If x and y are both positive odd integers then prove that x^2+y^2 is an even integers but not divisible by 4

Answer 1

let x = 2m+1, y= 2n+1 ( where x and y are positive integer )

=

$x {}^{2} + y {}^{2} = (2m + 1) {}^{2} + (2n + 1) { }^{2}$

$= 4{(m {}^{2} + n {}^{2} ) +( m + \: n)} + 2$

$4q + 2$

$where \: q \: = (m {}^{2} + \: n {}^{2} ) + (m + n)$

$x {}^{2} + y {}^{2} \: is \: even \: integer \: but \: not \: divisible \: by \: 4$

Answer 2

Answer: 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

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