# All the binomial coefficients $\binom{n}{ i}$ are divisible by a prime $p$ only if $n$ is a power of $p$.

I'm looking for a "high school / undergraduate" demonstration for the:

All the binomial coefficients $\binom{n}{i}=\frac{n!}{i!\cdot (n-i)!}$ for all i, $0\lt i \lt n$, are divisible by a prime $p$ only if $n$ is a power of $p.$

There is a "elementary" (good for high school) demonstration of reciprocal here: http://mathhelpforum.com/number-theory/186439-binomial-coefficient.html

And for this part I'm trying to avoid using more advanced tools like the Theorem of Lucas or Kummer.

Thanks in advance to anyone who can help.

• Do I understand correctly that you're looking for a combinatorial demonstration, rather than a proof using the expression for $\binom{n}{i}$? – whosleon Jun 24 '14 at 0:12
• Hello Whosleon! No problems in using the expression for $\binom{n}{i}$. But I'm searching for an "easy understanding" demonstration for students of high school. – Alex Jun 24 '14 at 0:46
• Sorry, but I have trouble understanding why this is true. Surely $3\mid{10\choose 3}=120$ and $5\mid{26\choose 5}=65780$? – Ian Mateus Jun 24 '14 at 0:58
• Hi Ian! Note $0\lt i \lt n$, and 3 not divide $\binom{10}{9} = 10$. – Alex Jun 24 '14 at 1:10
• Ian now I understood you. I improved the text. Thanks. – Alex Jun 24 '14 at 1:13

Let $a=p^nk$. Then let $p^m$ be the greatest power of $p$ smaller than or equal to $p^nk$. This number must be strictly smaller than $p^nk$.
Look at $$\binom{p^nk}{p^m}=\frac{ p^nk\cdot(p^nk-1)\cdot(p^nk-2)\dots(p^nk-(p^nk-p^m+1)}{p^m\cdot (p^m-1)\dots 1}$$ The factors on the top form a complete residue class mod $p^m$, as do the elements in the bottom. Therefore we can pair them up so they are congruent mod $p^m$. But since no number less than or equal to $p^nk$ is divisible by $p^ {m+1}$ this gives us all the information to determine the maximum power of $p$ that divides both the top and the bottom, and it is the same. Therefore that fraction is not divisible by $p$.
• Hello Bananarama, sorry but I do not understand why you used $i=p^m$ – Alex Jun 24 '14 at 0:49
• I use $p^m$ because if $a$ is not a power of $p$ then $\binom{a}{p^m}$ is not a multiple of $p$ – Jorge Fernández Hidalgo Jun 24 '14 at 0:51
• Look at the factors on the top and bottom, both are a complete residue class mod $p^m$, notice $p^m$ is the highest power of $p$ between $1$ and $a$. Therefore this tells us the product on top is divisible by $p$ the same amount of times as the product on the bottom, meaning the fraction is not dividible by $p$: – Jorge Fernández Hidalgo Jun 24 '14 at 0:58