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I'll present the explanation in a visual way, since people often like that, but the logic of this explanation could just as well be stated in non-geometric terms.

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It'll eventually get to the (36, 60) we constructed this line to go through, and thus (36, 60) itself is a multiple of (3, 5). Every grid point on the line is a multiple of the smallest one.

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(This is called "Euclid's lemma", but names don't matter)

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This was not an obvious tautology, primitive and unquestionable. It's a fact with an explanation, a story in answer to questioning it. It gets explained through a story like the above.

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If you multiply together a bunch of primes that aren't P, then the result won't be divisible by P. The only way to get a result divisible by P is for one of the primes to be P, by what we just showed.

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The End.

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But it's a little more abstract feeling than thinking of lines and grid points, so I went with the latter for this.

The End, for real.

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