Question

Prove that if a is an integer other than\(0\), then

a) \(1\)divides a.

b) a divides\({\rm{0}}\).

Short answer

(a) \(1\)divides a and,

(b) a divides \(0\).

Step-by-step solution

Step 1

DEFINITIONS

a divides b if there exists an integer c such that \(ac = b\)

Notation: \(a|b\)

Step 2

SOLUTION

(a) Given: a is an integer other than \(0\).

To proof: \(1\)divides a.

PROOF

Let \(c = a\).

c is an integer, because a is an integer. Using the identity property of multiplication.

\(1 \cdot c = 1 \cdot a = a\)

By the definition of divides, we have then shown that \(1\)divides a.

Step 3

(b) Given: a is an integer other than \(0\).

To proof: a divides\(0\).

PROOF

Let \(c = 0\).

Note that c is an integer, because \(0\)is an integer. Using the zero property of multiplication:

\(a \cdot c = a \cdot 0 = 0\)

By the definition of divides, we have then shown that a divides \(0\).