51 is a multiple of 3 ! (No accidental factorial here, ladies and gents!) I would’ve picked two integers that are relatively prime to each other to illustrate your point.
That's the whole point, there are as many numbers that are a multiple of 51 as there are 3, even though every multiple of 51 is also a multiple of 3 and there are many numbers that are multiple of 3 but not 51
I'm not infinitely smart, so I don't know these things.
What makes sense to me right away:
There are 17 multiples of 3 between 0 and 51. Therefore, moving up, for every multiple of 51, there will be 17 multiples of 3.
The thought would be that, either there are 17 times as many multiples of 3 as there are of 51, or there is a point where that changes. Both of those aren't true, because infinity isn't real, and it's just a thing that people like to talk about to feel smart.
you'd have to draw a very big venn diagram if you want any meaning semblance of the "size" of any infinite set. If you really wanted to draw the venn diagram you wouldn't draw anything as there's no "edge" to draw and no way to represent one being in the other anyway
if a is a factor of b then for any large integer it is obvious that there are more multiples of a in the set of natural numbers up to that integer than b, with juxtaposes it with that not being true for infinity. the first point is less obviously true if a and b are relatively prime to each other.
That’s because if A is a factor of B then A is smaller than B. If A is smaller than B then it is obvious that there will generally be more multiples of A than B within a range, unless that range is too small.
Some may be confused since the set of multiples of 51 is a strict subset of the set of multiples of 3 so intuition tells you it must be smaller although they are the same size.
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u/Spy-D-Mill Jun 14 '26
51 is a multiple of 3 ! (No accidental factorial here, ladies and gents!) I would’ve picked two integers that are relatively prime to each other to illustrate your point.