2. For , prove that ^{n} = 1 if and only if 3 | n, for any integer n. Solution
3. Solve the congruence 24x 168 (mod 200). Solution
4. Solve the following system of congruences.
2x 9 (mod 15) x 8 (mod 11).
Solution5. List the elements of Z_{15}^{×}. For each element, find its multiplicative inverse, and find its multiplicative order. Solution
6. Prove that if n > 1 is an odd integer, then (2n) = (n). Solution
7.
Prove that 5^{2n} - 1 is divisible by 24,
for all positive integers n.
(a) Give a proof using mathematical induction.
(b) Give a proof using congruences.
Solution
8. Prove that n^{5} - n is divisible by 30, for all integers n. Solution
Solutions to the problems | Forward to §2.1 | Back to §1.4 | Up | Table of Contents