Question

Prove that 4 is a factor of (5^n) + 3 for all natural integers n.

Answer 1

P(n)=5^n+3
P(1)=5^1+3=8
which is divisible by 4
so P(1) is true.
Assume that P(k) is true.
5^k+3 is divisible by 4
5^k+3=4m,where m is an integer.
5^k=4m-3
\[P \left( k+1 \right)=5^{k+1}+3=5^k*5^1+3=\left( 4m-3 \right)5+3=20m-15+3=20m-12\]
=4(5m-3)
Hence P(k+1) is divisible by 4
By induction P(n) is divisible by 4
or 4 is a factor of P(n)