Question

Prove or disprove that if \(a \mid b c\), where \(a, b\), and \(c\) are positive integers and \(a \neq 0\), then \(a \mid b\) or \(a \mid c\).

Short answer

disproven

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 positive integers with \(a\mid bc\) and \(a \ne 0\)

Counterexample

4 divides 36, because \(36 = 4 \cdot 9\) with 9 an integer.

However, \(36 = 6 \cdot 6\) and 4 does not divide 6 \({\rm{\;(as\;}}6 = 4 + 2{\rm{\;with\;}}0 < 2 < 4{\rm{\;)}}{\rm{.\;}}\)

Thus the statement is then false for \(a = 4\),\(b = c = 6\).