Question

show that square of any positive integer is in the form of 5q ,5q + 1,5 q+ 4 for some integer q
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Answer 1

Any positive integer n is off the firm 5 m or 5m+1 or 5m+2 or 5m+3 or 5m+4

If n=5/,then

n²=25m²=5(5m²)=5q where q=5m²

If n=5n+1,then

n²=(5m+1)²=5m(5m+2)+1=5q+1 where q=m(5m+2)

If n=5m+2,then

n²=(5m+2)²=5m(5m+4)+4=5q+4 where q=m(5m+4)

If n=5m+3,then

n²=(5m+3)²=5(m²+6m+1)+4=5q+4 where q=5m²+6m+1

If n=5m+4 then

n²≠5(5m²+8m+3)+3=5q+1 where q=5m²+8m+3

Hence n² is of the form 5q or 5q+1 or 5q+4