## anonymous one year ago Prove that 4 is a factor of (5^n) + 3 for all natural integers n.

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)