prove by mathematical induction that 7^n+4^n+1 is divisible by 6

Answer 1

7^n + 4^n + 1

1. test for n = 1
7^1 + 4^1 + 1 = 12 , which is divisible by 6

2. assume it is true for n = k , that is, assume that
7^k + 4^k + 1 is divisible by 6

3. then show that 7^(k+1) + 4^(k+1) - 1 is divisible by 6

use the number property that if both A and B are divisble by c
then A-B is divisible by c
e.g. 156 and 117 are both divisible by 13
then is 156-117 or 39 divisible by 13 ? YES

so ...
7^(k+1) + 4^(k+1) + 1 - (7^k + 4^k + 1)
= 7^(k+1) + 4^(k+1) + 1 -7^k - 4^k - 1
= 7^k(7-1) + 4^k(4-1)
= 6(7^k) + 3(4^k)

clearl 6(7^k) is a multiple of 6 , thus divisible by 6
and in 3(4^k) , the 4^k must be even and any even times 3 is divisible by 6
so we have shown that the result is divisible by 6
(the sum of multiples of 6 must be divisible by 6)