Let \(a\text{,}\) \(b\text{,}\) and \(c\) be integers. Assume that \(a|b\) and \(b|c\text{.}\) In other words, \(b\) is a multiple of \(a\) and \(c\) is a multiple of \(b\text{.}\) So there are integers \(k\) and \(j\) such that \(b = ka\) and \(c = jb\text{.}\) Combining these (through substitution) we get that \(c = jka\text{.}\) But \(jk\) is an integer, so this says that \(c\) is a multiple of \(a\text{.}\) Therefore \(a|c\text{.}\)
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