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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.

glimpser
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    It's exactly that. The problem of finding all multiples of $n$ is trivial: we just have ${mn:m\in\Bbb Z}$. – pancini Apr 21 '20 at 02:33
  • There are some topics involving "multiple," such as "least common multiple." – P. Lawrence Apr 21 '20 at 02:36
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    @ElliotG It's not necessarily trivial, e.g. the involution (reflection) $, n\mapsto ab/n,$ gives an $\rm\color{#c00}{order\ reversing}$ bijection of common divisors of $,a,b,$ with common multiples of $a,b$ (below $ab),,$ so it maps the $\rm\color{#c00}{greatest}$ common divisor to the $\rm\color{#c00}{least}$ common multiple thus $,{\rm lcm}(a,b)= ab/\gcd(a,b).,$ Follow the link for more on such duality. – Bill Dubuque Apr 21 '20 at 03:54

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