1. Establish the values of k for which the binomial coe?cient p Ck is divisible by p when p is a prime number. Use your result and the method of induction to prove that np - n is divisible by p for all integers n and all prime numbers p. Deduce that n5 - n is divisible by 30 for any integer n.

2. An arithmetic progression of integers an is one in which an = a0 + nd, where a0 and d are integers and n takes successive values 0, 1, 2, ... .

(a) Show that if any one term of the progression is the cube of an integer then so are in?nitely many others.

(b) Show that no cube of an integer can be expressed as 7n + 5 for some positive integer n.