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Chapter 1 Review: Solved problems

1. Find gcd (7605, 5733), and express it as a linear combination of 7605 and 5733.     Solution

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).

Solution

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


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