Chapter 1

Question 1

Even

Answer 1

An integer is called even provided it is divisible by two.. by definition of divisible, 2|x means there is an integer a where x = 2a.

Question 2

Divisible

Answer 2

Let a and b be integers. We say that a is divisible by b provided there is an integer c such that bc = a. We also say b divides a, or b is a factor of a, or b is a divisor of a. The notation for this is b|a.

Question 3

Odd

Answer 3

An integer a is called odd provided there is an integer x such that a = 2x + 1.

Question 4

Prime

Answer 4

An integer p is called prime provided that p > 1 and the only positive divisors of
p are 1 and p.

Question 5

Composite

Answer 5

A positive integer a is called composite provided there is an integer b such that
1 < b < a and b|a.

Question 6

Vacuous Truth

Answer 6

Statements of the form “If A, then B” in which condition A is impossible are called vacuous, and mathematicians consider such statements true because they have no exceptions.

Question 7

Proposition 5.2

Answer 7

The sum of two even integers is even.

Question 8

Proposition 5.3

Answer 8

Let a, b, and c be integers. If a|b and b|c, then a|c.