Is there a positive integer $n$ where $14$ divide $30^n$? Explain why.

Question

The lecturer has not taught us proofs yet, so I think the question is not looking for a rigorous proof. My attempt:

$$30 = 2 \times 3 \times 5$$ $$\frac{30^n}{14} = \frac{2^n \times 3^n \times 5^n}{2 \times 7} = \frac{2^{n-1} \times 3^n \times 5^n}{7}$$

Now I have written this in prime numbers, which I presume will make it easier to solve.
I feel that $7$ is not going to divide any power of $2$, $3$, or $5$.
This is as close as I can get to a proof.

Answer 1

Your idea is fine, but you don't have to actually divide $30^n$ by $14$. Just note that $30^n=2^n3^n5^n$ (which you did) and that therefore, since there is no $7$ here (and since $7$ is prime), $2^n3^n5^n$ is not a multiple of $7$. In particular, it is not a multiple of $14$.

Answer 2

We have $$30^n\equiv 2,4,8\mod 14$$ so exists no such exponent for $$n\geq1 $$ and $$n\in\mathbb{N}$$

Answer 3

You can also reason as follows:

• $30^n = (2\cdot 14 + 2)^n = m\cdot 14 + 2^n$ for a positive integer $m$
• It follows if $14 | 30^n$, then $14| 2^n$ which is impossible as $7$ is not a factor of any power $2^n$