2007 IMO Shortlist N7

For a prime $p$ and a positive integer $n$, denote by $\nu_{p}(n)$ the exponent of $p$ in the prime factorization of $n$ !. Given a positive integer $d$ and a finite set $\left\{p_{1}, \ldots, p_{k}\right\}$ of primes. Show that there are infinitely many positive integers $n$ such that $d \mid \nu_{p_{i}}(n)$ for all $1 \leq i \leq k$. (India)

对于质数 $p$ 和正整数 $n$，用 $\nu_{p}(n)$ 表示 $p$ 在 $n$ ! 的质因数分解中的指数。给定一个正整数 $d$ 和质数的有限集 $\left\{p_{1}, \ldots, p_{k}\right\}$。证明存在无限多个正整数 $n$，使得所有 $1 \leq i \leq k$ 都满足 $d \mid \nu_{p_{i}}(n)$。 (印度)