Example

Suppose 0 < p^k | m for some prime number p and integer k.

Suppose k is maximal with this property. Then the binomial coefficient

m pk

is not divisible by p.

Indeed, this binomial can be written as the quotient of

m · (m - 1) ··· (m - p^k + 1) by

p^k!

Now for all 0 n p^k we have

ord_p(m - n) = min(k,ord_p(n)) = ord_p(p^k - n),

and every factor p in the numerator is canceled by a factor p in the denominator.