Question

Prove that part\((ii)\)of Theorem\(1\)is true.

Short answer

If \(a|b\), then \(a|bc\)for all integers c.

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

Given: \(a|b\)

To proof: For every integer \(c:a|bc\)

PROOF

Let c be an integer.

Since \(a|b\), there exists an integer d such that

\(a \cdot d = b\)

Multiply each side of the previous equation by c:

\(a \cdot d \cdot c = b \cdot c\)

Use associative property of multiplication:

\(a \cdot (dc) = bc\)

Since d and c are both integers, their product dc is also an integer.

By the definition of divides, we have then shown that a divides bc.

\(a|bc\).