2014 Putnam B3

Let $A$ be an $m \times n$ matrix with rational entries.

Suppose that there are at least $m+n$ distinct prime numbers among the absolute values of the entries of $A$. Show that the rank of $A$ is at least 2.

令 $A$ 为具有有理项的 $m \times n$ 矩阵。

假设$A$ 的条目的绝对值中至少有$m+n$ 个不同的素数。证明 $A$ 的等级至少为 2。