Definition: We say that one integer a divides a second integer b, and write a|b, if there is a third integer q such that b = aq.
As is often the case with precise mathematical definitions like this one, you have to poke around the edges for a while before you really understand what it's trying to say. Definitions, like fences, serve both to keep certain things in and other things out. Everyone who has survived the standard mathematics courses in elementary school has confronted the rigors of long division, and consequently is well aware that you can divide anything (except 0) into anything else, provided you are willing to suffer the slings and arrows of quotients and remainders. The thing that is being fenced out by the present definition of divides is remainders. We won't allow them. Remainders are forbidden. So, 3 divides 6, but 3 does not divide 7.
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