Riley hobson bence · Problem 1.27

Preliminary algebra

Mathematics — Methods · binomial-coefficients-divisibility-induction

Problem

Establish the values of

k

for which the binomial coefficient

{}^pC_k

is divisible by

p

when

p

is a prime number. Use your result and the method of induction to prove that

n^p - n

is divisible by

p

for all integers

n

and all prime numbers

p

. Deduce that

n^5 - n

is divisible by 30 for any integer

n