Proof for positive integer

Question

Prove that for any positive integers $m$ and $n$, there exists a set of $n$ consecutive positive integers each of which is divisible by a number of the form $d^m$ where $d$ is some integer in $\mathbb{N}$.

Answer 1

Hint

Let $p_1,p_2,..,p_n$ be pairwise distinct primes. By the Chinese Remainder Theorem, the system $$x \equiv 0 \pmod{p_1^m} \\ x+1 \equiv 0 \pmod{p_2^m} \\ .... \\ x+n-1 \equiv 0 \pmod{p_n^m}$$ has solutions.