Question

For any positive integer $n$, the expression $6^n-1$ is divisible by

• A. 2
• B. 3
• C. 5
• D. 7

Answer 1

The correct option is C 5

If we check the options we find that only 5 can divide the expression for $n=1$. To be sure if the expression is divisible by 5 for any value of n we will have to use the principle of mathematical induction.
So, let for any $n\ge 1$, let $P_n$ be the statement that $6^n-1$ is divisible by 5.

$\underset{–}{BaseCase}$. The statement $P_1$ says that

$6^1$ − 1 = 6 − 1 = 5

is divisible by 5, which is true.

$\underset{–}{InductiveStep}$. Fix $k\ge 1$, and suppose that $P_k$ holds, that is, $6^k-1$ is divisible by 5.

It remains to show that $P_{k+1}$ holds, that is, that $6^{k+1}-1$ is divisible by 5.

$6^{k+1}-1=6\left(6^k\right)-1$

= $6(6^k-1)-1+6$

= $6(6^k-1)+5$.

By $P_k$, the first term $6(6^k-1)$ is divisible by 5, the second term is clearly divisible by 5. Therefore the left hand side is also divisible by 5. Therefore $P_{k+1}$ holds.

Thus by the principle of mathematical induction, for all $n\ge 1$, $P_n$ holds.