Prove that square of any integer leaves the remainder either 0 or… - Vidyadip

Question

Prove that square of any integer leaves the remainder either 0 or 1 when divided by 4

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Answer

Let the integer be $x$
The square of its integer is $x^2$
Let $x$ be an even integer
$x=2 q+0$
$x^2=4 q^2$
When $x$ is an odd integer
$x=2 k+1$
$x^2=(2 k+1)^2$
$=4 k^2+4 k+1$
$=4 k(k+1)+1$
$=4 q+1 \ldots \ldots .[q=k(k+1)]$
It is divisible by 4
Hence it is proved

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