Smallest positive integer $n$ such that $2^{1980}\cdot3^{384}\cdot5^{1694}\cdot7^{343}\cdot n$ is a $k$-th power for some $k\ge2$.

Question

Let $n$ be the smallest positive integer such that $mn$ is a perfect $k$th power of an integer for some $k \ge 2$, where $m=2^{1980} \cdot 3^{384} \cdot 5^{1694} \cdot 7^{343}$. What is $n+k$?

I'm not sure what this problem is asking and what I need to do. Any help is greatly appreciated.

Answer 1

Note that if $n=2\times 3$, we will get, $$mn = (2^{1981})(3^{385})(5^{1694})(7^{343})$$ $$mn =(2^{283})^7(3^{55})^7(5^{242})^7(7^{49})^7$$ $$mn =(2^{283}\times 3^{55}\times 5^{242}\times 7^{49})^7$$ Thus, here $n=6$ and $k=7$, giving $n+k=13$.