Is the expression $\frac{\gcd(m,n)}{n}\binom{n}{m}$ always an integer?

[Ackbach]

Here is this week's POTW:

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Prove that the expression
\[ \frac{\gcd(m,n)}{n}\binom{n}{m} \]
is an integer for all pairs of integers $n\geq m\geq 1$.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[Ackbach]

Re: Problem Of The Week # 266 - Jun 05, 2017

This was Problem B-2 in the 2000 William Lowell Putnam Mathematical Competition.

Congratulations to joypav for his correct solution, which follows:

Spoiler