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

or

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.

More later,

Nalin Pithwa.

19.209537
72.864102

Thakur Complex, Kandivali East, Mumbai, Maharashtra 400101, India

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