Question

Prove that the square of any positive integer in of the from 5 to 5 cube + 152 + 4 of same integer q

Answer 1

Answer:

Let x be any positive integer

then, x=5q or x=5q+1 or x=5q+4 for integer x

If x=5q

x

2

=(5q)

2

=25q

2

=5(5q

2

)=5n

where n=5q

2

If x=5q+1

x

2

=(5q+1)

2

=25q

2

+10q+1=5(5q

2

+2q)+1=5n+1

where n=5q

2

+2q

If x=5q+4

x

2

=(5q+4)

2

=25q

2

+40q+16=5(5q

2

+8q+3)+1=5n+1

where n=5q

2

+8q+3

∴ In each three cases x

2

is either of the form 5q or 5q+1 or 5q+4 and for integer q.