Assignment 3

Question 1

Have you read Sections 1.2 and 1.3? Circle one: Yes / No.
(No credit for an answer of “No”.)

Question 2

State the Fundamental Theorem of Arithmetic.

Question 3

If \( p > 3 \) is prime, prove that \( p^2 + 2 \) is composite.
Hint: Use the Division Algorithm to divide \( p \) by 3.

Question 4

If \( a, b, c \) are integers and \( p \) is a prime that divides both \( a \) and \( a + bc \), prove that \( p \mid b \) or \( p \mid c \).

Question 5

State the definition of what it means for a positive integer \( p \) to be prime, and write the equivalent condition given in Theorem 1.5. A third equivalent condition is given by the following two exercises.

Question 6

Suppose \( p \) is prime. Prove that for each \( a \in \mathbb{Z} \), either \( (a, p) = 1 \) or \( p \mid a \).

Question 7

Suppose that \( p \) has the following property: For each \( a \in \mathbb{Z} \), either \( (a, p) = 1 \) or \( p \mid a \). Prove that \( p \) is prime.

Question 8

The integer \( p = 6 \) is not prime; find an integer \( a \) such that \( (a, p) \neq 1 \) and \( p \nmid a \).

Question 9

Suppose \( p \) is a positive prime and \( n \geq 2 \) is an integer. Prove that \( \sqrt[n]{p} \) is irrational.