prove that the square of any positive integer is of form 4w , 4w+1 , 4w+2 or 4w+3 for ssome integer w.

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Solution

let the positive integer be 4p, 4p+1, 4p+2
If n= 4p
n²=16p²
=4(4p²)
=4w
Similarly,
If n= 4p+1
n²=(4p+1)²
=16p²+8p+1
= 4(4p²+4p)+1
=4w+1
If n=4p+2
n²=(4p+2)²
=16p²+16p+4
=4(4p²+4p+1)
=4w
Hence proved.

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