Divisibility

Saying \(a\mid b\) is the same as saying there is an integer \(k\) with \(b=ak\); no division or decimal result is involved. Thus \(3\mid 12\) (with \(k=4\)) and \(5\nmid 12\). Every integer divides 0, because \(0=a\cdot0\), and 1 and \(-1\) divide everything; zero, however, divides nothing but itself, which is why \(a\neq0\) is required.

The basic properties follow directly from the definition. The relation is reflexive (\(a\mid a\)) and transitive (if \(a\mid b\) and \(b\mid c\), then \(a\mid c\)), but not symmetric: \(2\mid 6\) and \(6\nmid 2\). If \(a\) divides \(b\) and \(c\), it divides every combination \(mb+nc\); this observation underlies the Euclidean algorithm.

When \(a\nmid b\), division with remainder still makes sense: there are unique integers \(q\) and \(r\) with \(b=aq+r\) and \(0\le r<a\), and \(a\mid b\) is equivalent to \(r=0\). A common mistake is to think that \(a\mid bc\) forces \(a\mid b\) or \(a\mid c\): \(4\mid 2\cdot2\), yet 4 does not divide 2. That implication is valid only when \(a\) is prime.

Divisibility leads to the greatest common divisor, to prime numbers and to modular arithmetic, where \(n\mid a-b\) reads as «\(a\) and \(b\) are congruent modulo \(n\)».