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