Exercise 1531Combinatorics › Binomial coefficientsOlympiadAdded 3 Nov 2026

Binomial coefficients and primes

#prime#binomial#divisibility

Statement

Let $p$ be prime. Show that $p$ divides $\binom{p}{k}$ for $1\le k\le p-1$ .

Hint

Use $\binom{p}{k}=\frac{p!}{k!(p-k)!}=\frac{p\cdot(p-1)!}{k!(p-k)!}$ .

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Rewrite. $\binom{p}{k}=\frac{p}{k}\cdot\binom{p-1}{k-1}$ .

Integrality. $\binom{p}{k}$ is an integer. Since $p$ is prime and $1\le k\le p-1$ , $\gcd(p,k)=1$ . So $p\mid\binom{p}{k}$ .

Conclusion. $p$ divides all inner binomial coefficients in row $p$ .

Application. $(a+b)^p\equiv a^p+b^p\pmod p$ (Freshman's dream).