# Prove that the square of any positive integer is of the form 5q or,5q+1 or,5q+4 for some integer q.

1
by srishtu

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by srishtu

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(5m)² = 25m² = 5 X 5m² = 5q (where q = 5m²)

(5m+1)² = (5m)² + 2X 5m X 1 + 1² (by using identity- (a+b)²= a²+ 2ab +b² )

= 25m² + 10 m + 1

= 5 (5m²+ 2m) +1

= 5q +1 ( q= 5m²+2m)

(5m+2)²= (5m)² + 2X 5m X 2 +2² (by using identity- (a+b)²= a²+ 2ab +b² )

= 25m² + 20 m +4

= 5 (5m²+4m ) + 4

= 5q+4 (where q= 5m²+4m)

Hence proved :) :D