1(a) Find the remainder when is divided by 17.
1(b) Find the remainder when is divided by 29.
2: Determine whether 17 is a prime by deciding if
3: Arrange the integers 2,3,4, …, 21 in pairs a and b that satisfy .
4: Show that .
5a: Prove that an integer is prime if and only if .
5b: If n is a composite integer, show that , except when .
6: Given a prime number p, establish the congruence
7: If p is prime, prove that for any integer a, and
8: Find two odd primes for which the congruence holds.
9: Using Wilson’s theorem, prove that for any odd prime p:
10a: For a prime p of the form , prove that either
10b: Use the part (a) to show that if is prime, then the product of all the even integers less than p is congruent modulo p to either 1 or -1.