Question

For all $n\in N$, which of the following is a factor of $2^{3n}-1$?

• A. 3
• B. 5
• C. 7
• D. none of these

Answer 1

The correct option is C 7

We can see that when $n=1$, the expression $2^{3\times 1}-1=2^3-1=8-1=7$
Similarly, when $n=2$, the expression $2^{3\times 2}-1=2^6-1=64-1=63$
And when $n=3$, the expression $2^{3\times 3}-1=2^9-1=512-1=511$
All of these values we got are divisible by $7$.
Hence, we can conclude that for all natural numbers, $2^{3n}-1$ is divisible by $7$.