= Lucas's theorem
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{title2=$\binom nk\equiv\prod_i\binom{n_i}{k_i}\pmod p$}
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= Lucas theorem
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{synonym}
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 $\binom ab=0$ when $b>a$. The identity follows by factoring $(1+x)^n$ into digit powers and using the <Frobenius endomorphism>; digit uniqueness identifies each coefficient.
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