Chapter 1, Problem 17P

To determine

To prove: If n is positive integer, then $7^n-1$ is divisible by 6.

Explanation of Solution

Proof:

Use mathematical induction on n.

Let $S_n$ be the statement $7^n-1$ is divisible by 6

Base case: $n=1$

To prove that the statement is true for $n=1$.

$7^1-1=6$

Clearly, it is divisible by 6.

Therefore, the statement is true for $n=1$.

Induction hypothesis: $n=k$

Assume that the claim is true for $n=k$.

That is, $7^k-1=6m\text{, for some positive integer }m$

Thus $7^k=6m+1$.

Inductive step: $n=k+1$

To prove that the statement is true for $n=k+1$.

$\begin{array}{c}{7}^{k+1}-1=7^k\cdot 7-1\\ =\left(6m+1\right)\cdot 7-1\left[\text{From induction hypothesis}\right]\\ =42m+6\\ =6\left(7m+1\right)\end{array}$

Thus, the statement is true for $n=k+1$.

Therefore, by mathematical induction, the statement $7^n-1$ is divisible by 6.

Hence the required result is obtained.