number theory for finding value of $k$

Question

How do I find what is the smallest positive integer $k$ such that $(3^3 + 4^3 + 5^3)\cdot k = a^n$ for some positive integers $a$ and $n$, with $n > 1$?

Answer 1

You evaluate $3^3+4^3+5^3$ and factor it. Then you think about what factors need to be increased so they are all multiples of some number.

Answer 2

Since $3^3+4^3+5^3=6^3$, we have $$ 6^3k=a^n $$ Thus $2|a$ and $3|a$, so $6|a$.

Now for $n=1$, $k=1$ suffices, and it is easy to see that for $n=2$, $k=6$ is good, and no smaller number suffices. If $n=3$, then $k=1$ is good as well. Otherwise, $a\geq 6\Rightarrow$ $a^n\geq 6^n$ and thus $k\geq 6^{n-3}$. But $k=6^{n-3}$ is a good choice.

Answer 3

$(3^3 + 4^3 + 5^3)\cdot k =6^3k= a^n $ in which is obvious that $(k,n,a)=(1,3,6)$