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

Reiny
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 AB is divisible by c
e.g. 156 and 117 are both divisible by 13
then is 156117 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(71) + 4^k(41)
= 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)
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