Why n and what does it mean?

Question

Maybe a bit too early to ask, but I give it a shot. I'm trying to study mathematics again during my free time. Now I tried to figure out what exactly is meant with the multiple. Well this formula makes it quite easy to understand for me, but it's not totally clear yet:

a = n x b

So if I understand the formula correctly, for 15 = 3 * 5 15 is a multiple of 5, and for 15 = 5 * 3 15 is a multiple of 3, so 3 and 5 can both be a multiple of 15. But what I don't understand, why n? Why not just a = b x c? I cannot really find an answer for this either, in my study book they just start with formulas like:

aⁿ x a ^m = a ^n+m

without any further explanation why to use n or m. Hopefully somebody has an explanation for me

Answer 1

Why not? Any letter or symbols work. You could have emoticons such that

Sad = happy x heart

If it suits, the usage of letters is due to their ease of writing. This comes from the intense repitition of them through the years. So which letter you pick does not matter, why n? The author felt like it?

Answer 2

The author probably wanted to stress that he was talking about integer multiples of $b$. $n$ is commonly used to denote natural numbers. Of course, $a$ and $b$ are also naturals here, but the formula could also work with $a,b$ rational or real.

It is also pretty common to work with letters in the same alphabetic range to denote a similar role/type. For instance, a polynomial will have coefficients $a,b,c,d,\cdots$. For the same reason, $i,j,k,l,m,n$ often denote integers (indexes or counts). $x,y,z,t,u,v,w$ often denote unknowns or variables.