Show that for any integer n22 is not divisble by 4 n geq 0 S

Show that for any integer n^2+2 is not divisble by 4. n geq 0.

Solution

First let us assume n =1

Obviously 5 is not divisible by 4

Induction method is going to be used now

Assume k^2+1 is not divisible by 4

consider (k+1)^2+1

=k^2+2k+2

As k^2 is not divisible by 4,

we have k is not divisible by 2.

Hence k^2 is not divisible by 4 and 2k not divisible by 4.

Thus the sum k^2+2k+2 is not divisible by 4.

Hence true for k+1.

Thus proved by induction