I'm currently trying to learn about totient, while following the proof of the fermat's little theorem I got stuck at some part and that part include a question of title.
first, to prevent the confuse of word, I'll fix the meaning of specific word here.
order: the number of multiplication takes to circle around and get back to where it started at modular multiplicative cycle.
and for simplicity, I want further argue to be based on assumption that the modular multiplicative cycle I'll mention afterward will be multiple of a (mod m)
Although the question is already stated at title, Why the order of modular multiplicative cycle doesn't change when entire cycle is multiplied by number which is relatively prime to m
in other word, Why the order of modular multiplicative cycle changes when entire cycle is multiplied by number not relatively prime to m?
p.s. If you insist me specific part confusing to you, then i'll try my best to correct the word to be more based on definition and convention.
A basic fact about cyclic groups is that $\mid g^k\mid=\frac n{(n,k)}$, where $n=\mid g\mid$.
Thus the order changes iff $k$ is not relatively prime to $n$. (Of course, this is multiplicative notation. For additive notation, we get $\mid kg\mid=\frac n{(n,k)}$).