So far as I've been reading any number theory and algebra books,
They seem to take more care of "divisor" than of "multiple".
But I haven't seen any kind of number theory beginning with "multiple".
I guess one of the reasons is that multiples are easy to be calculated, you just multiply. (Of course, it takes a little bit of time.)
But divisors are not easy, if you give me tremendously big number, say $10^{100}$-digit number and ask me to find its divisors, I couldn't do that even if I'm allowed to use my calculator.
I guess there is a huge gap between their complexity of computations.
I'd like to know whether there is any reason why they start with divisors.