If x and y are two odd number then prove that x^2 + y^2 is not divisible by 4.
In equation form.
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x and y are odd numbers
let x = 2n +1
y = 2m + 1
where m , n are integers
now,
x² + y² = (2n +1)² +(2m +1)²
= 4n² + 4n + 1 + 4m² + 4m +1
= 4n² + 4m² + 4m + 4n +2
= 4{ m² + n² + m + n } + 2
if divide this by 4
we see that
4( m² + n² + m + n )/4 + 2/4
last term e.g 2/4 = 1/2 is fraction
hence, x² + y² isn't divisible by 4 when x and y are odd numbers.
let x = 2n +1
y = 2m + 1
where m , n are integers
now,
x² + y² = (2n +1)² +(2m +1)²
= 4n² + 4n + 1 + 4m² + 4m +1
= 4n² + 4m² + 4m + 4n +2
= 4{ m² + n² + m + n } + 2
if divide this by 4
we see that
4( m² + n² + m + n )/4 + 2/4
last term e.g 2/4 = 1/2 is fraction
hence, x² + y² isn't divisible by 4 when x and y are odd numbers.
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