Chapter 1 Review: Solved problems

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).

5. List the elements of Z₁₅^×. 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 5²ⁿ - 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 n⁵ - 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