Prove $n \choose m$ is divisible by n when $\gcd(m,n)=1$

Question

I know that $p \choose k$ is divisible by $p$ for all $k \in \{1, 2, \ldots,p-1\}$. But I am trying to work on a generalization as in the title but do not know how to prove it.

Please give more context. Providing context not only assures that this is not simply copied from a homework assignment, but also allows answers to be better directed at where the problem lies and to be within the proper scope. Please avoid "I have no clue" questions. Defining keywords and trying a simpler, similar problem often helps. — robjohn, Sep 15 '19 at 19:27

score 3 · Answer 1 · answered Sep 15 '19 at 19:01

3

Hint: Consider the fact that $\binom{n-1}{m-1}$ is an integer, and $\binom nm$ is $\frac nm$ times larger.

answered Sep 15 '19 at 19:01

Arthur

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Prove $n \choose m$ is divisible by n when $\gcd(m,n)=1$

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