## Description

1. (25 points) Let p be a prime number and a an integer not divisible by p. Prove that

there exists an integer b such that ab ≡ 1 (mod p). Conclude that every non-zero

element in Zp has a multiplicative inverse.

Hint: Use Bezout’s identity.

2. (25 points) Let x be an integer such that x leaves a remainder 3 when divided by 5,

leaves a remainder 7 when divided by 12 and it leaves a remainder 8 when divided

by 13. Find the remainder when you divide x by 780 using the Euclidean algorithm.

Hint: Chinese remainder theorem and the example following it in textbook.

3. (25 points) Let n be a positive integer. Prove that if 2n − 1 is a prime number then

n is a prime number.

4. (25 points) Prove that there does not exist integers any a and b such that the function

p(n) = n

2 + an + b is a prime number for every positive integer n.

Hint: First prove that if such a polynomial exists then b = 1. Then use modular

arithmetic to conclude the result.

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