Binomial Coefficient modulus Prime?

Number Theory Level 5

Let $p$ be a prime, $k,n\in\mathbb N$ .
Then $\displaystyle\binom{p^nk}{p^n}\equiv \, ?(\text{mod}\, p)$

n-1

1

0

n

k

p-1

p-k

1 solution

展豪張
Mar 28, 2016

By lucas's theorem , $\displaystyle\binom{p^nk}{p^n}\equiv\binom{k}{1}=k(\text{mod}\,p)$