ILL 1972 Problem 32

If $n_1, n_2, \cdots, n_k$ are natural numbers and $n_1+n_2+\cdots+n_k = n$ , show that $$max(n_1n_2\cdots n_k)=(t + 1)^rt^{k-r},$$ where $t =\left[\frac{n}{k}\right]$ and $r$ is the remainder of $n$ upon division by $k$ ; i.e., $n = tk + r, 0 \le r \le k- 1$ .