Question

Prove that n2+n+1 is not divisible by 5 for any n?

Answer 1

let n=5q+r; 0 <_5<b
therefore five cases will be formed
n=5q
n=5q+1
n=5q+2
n=5q+3
n=5q+4
taking each case one by one
=(5q)^2+5q+1
=25q^2+5q+1
=5(5q^2+q)+1
[let 5q^2+q=m]
=5m+1
therefore it is not divisible by 5.
=(5q+1)^2+(5q+1)+1
=25q^2+10q+1+5q+1+1
=5(5q^2+2q+q)+3
[let 5q^2+2q+q=m]
=5m+3
therefore it is not divisible by 5.
=(5q+2)^2+(5q+2)+1
=25q^2+20q+4+5q+2+1
=25q^2+20q+5q+5+2
=5(5q^2+4q+q+1)
[let 5q^2+4q+q+1=m]
=5m+2
therefore it is not divisible by 5.
=(5q+3)^2+(5q+3)+1
=25q^2+30q+9+5q+3+1
=5(5q^2+6q+q+2)+3
[let 5q^2+6q+q+2=m]
=5m+3
therefore it is not divisible by 5.
=(5q+4)^2+(5q+4)+1
=25q^2+40q+16+5q+1
=25q^2+40q+15+1+5q+1
=5(5q^2+8q+q+3)+2
[let 5q^2+8q+q+3=m]
=5m+2
therefore it is not divisible by 5.
Therefore n^2+n+1 is not divisible by 5 for any value of n.