Lucas's theorem ($\left(\genfrac{}{}{0ex}{}{n}{k}\right)\equiv {\prod }_i\left(\genfrac{}{}{0ex}{}{n_i}{k_i}\right)(\mathrm{mod}p)$)

For a prime number $p$, the binomial coefficient modulo $p$ factors into the corresponding digit binomial coefficients in the base-p expansions of the nonnegative integers $n,k$. Pad both expansions to the same length and use $\left(\genfrac{}{}{0ex}{}{a}{b}\right)=0$ when $b>a$. The identity follows by factoring $(1+x{)}^n$ into digit powers and using the Frobenius endomorphism; digit uniqueness identifies each coefficient.