(related to Proposition: Finite Number of Divisors)

- If \(a \neq 0\) is an integer and $b$ its divisor \(b\mid a\), then, by definition, there exists an integer \(q\neq 0\) with \[a=qb.\]
- Since \(q\ge 1\), it follows from the properties of the absolute value and the rules of calculation with inequalities that $|a|=|q||b|\ge |b|.$
- Since \(|a|\) is finite and since \(|b| > 0\), there are only finitely many \(b\) such that \(|a|\ge |b|\).
- Thus, the number of divisors of \(|a|\) is also finite.∎

**Landau, Edmund**: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927