It is a well known rule, that $\frac{n}{\gcd(n,k)}$ divides $\binom{n}{k}$. Thus, $n$ divides the binomial coefficient if $\gcd(n,k) = 1$. But, what about the case: $n = 12$ and $k=6$? $12$ divides $\binom{12}{6} = 924$ even though $\gcd(12,6) = 6\neq 1$. What is the simple explanation to this "exception"?
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7"if" is not "only if". – darij grinberg Oct 19 '20 at 14:15
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Related: When is $\binom{2n}{n}\cdot \frac{1}{2n}$ an integer. – JMoravitz Oct 19 '20 at 14:18