# Minimum number of distinct entries to fill a matrix so that no two cols are the same

Let $A$ be an $m \times n$ matrix with entries from $\{1,2,...,k\}$. What is the minimum $k$ (as a function of $m$ and $n$) needed so that we can fill $A$ where no two columns are identical.

Thank you.

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Hint 1: If there are $m$ rows and $k$ symbols, how many possible distinct columns are there?
Hint 2: If the answer to hint 1 must be greater then or equal to the actual number of columns $n$, what does this mean that $k$ must be greater than or equal to?