TITLE: Can a multiple of $15$ and a multiple of $21$ differ by $1$?
QUESTION [5 upvotes]: I know a solution to this question having to do with the fact that the $\gcd(15, 21) = 3$, so the answer is no.
But I can't figure out what is the reasoning behind this. Any help would be really appreciated!

REPLY [11 votes]: It's really simple, imagine the two numbers as $15n$ and $21k$.
Suppose that it's possible, then: $15n-21k = 1$
As you said $\gcd(15, 21) = 3$, so you can factor out: $3(5n-7k) = 1$.
On the left side, you have a multiple of 3, and in the other you have a 1, which isn't, so you have a contradiction. Note that inverting the subtraction won't change anything.