Question

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

Short answer

If \(a|b\)and \(b|c\)then \(a|c\).

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: \(a|b\)and \(b|c\)

To proof: \(a|c\)

PROOF

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

\(a \cdot d = b\)

Since \(b|c\), there exists an integer f such that:

\(b \cdot f = c\)

Combining these two equations, we then obtain:

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

Since d and f are both integers, their product df is also an integer.

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

\(a|c\).