Question

Use the binomial theorem to show that 7^n + 2 is divisible by 3, where n is a positive integer.
(Hint: Write 7 = 6 + 1.)

Answer 1

By the binomial theorem,

$7^n=(6+1)^n=6^n+n6^{n-1}+\dbinom n26^{n-2}+\cdots+\dbinom n{n-2}6^2+n6^1+1$

Each term containing a factor 6 is divisible by 3, since 3 divides 6. The remaining 1 and 2 sum to 3, which is also divisible by 3.