Question

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

Short answer

\(ab\mid cd\)

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

To proof: \(ab\mid cd\)

PROOF

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

\(c = af\)

Since \(b\mid d\), there exists an integer \(g\) such that:

\(d = bg\)

Multiply these two equation :

\(cd = (af)(bg) = afbg = abfg = (ab)(fg)\)

Since \(f\) and \(g\) are integers, their product \(fg\) is an also integer .

By the definition of divides , we have then shown that \(ab\) divides \(cd\)

\(ab\mid cd\).