Question

Consider the number $6^n$, where n is a natural number. Check whether there is any value of n ∈ N for which $6^n$ is divisible by 7.

Answer 1

Solution:

Checking whether $6^n$ is divisible by $7$:

A number $a$ is divisible by a prime number $b$ only if the prime factorization of the given number $a$ contains the prime number $b$.

Since, $6=2\times 3$

$\therefore 6^n=2^n\times 3^n$

We can see that the prime number $7$ is not there in the prime factorization of $6^n$. So, it is not divisible by $7$.

Final answer: Hence, $6^n$ is not divisible by $7$ for any value of $n\in N$.