Question

Show that if \(a, b\), and \(c\) are integers, where \(a \neq 0\) and \(c \neq 0\), such that \(a c \mid b c\), then \(a \mid b\).

Short answer

\(a\mid b\)

Step-by-step solution

Step 1

DEFINITION

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

Notation : \(a\mid b\)

Step 2

SLOUTION

Given: \(a\),\(b\),\(C\) are integers with \(ac\mid bc\) and \(a \ne 0\) and \(c \ne 0\)

To proof: \(a\mid b\)

PROOF

Since \(a\mid c\) , there exists an integer \(d\) such that:

\(bc = (ac)d = acd\)

Since \(c \ne 0\), there exists an integer \(C\) such that:

\(b = ad\)

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

\(a\mid b\).