About The Order of an Integer

Question

In this bolg It says $x=ord_{n}b$ and $ord_n = $ the least positive integer x such that $b^x\equiv $ 1 (mod n) and below it says $b^x\equiv $ 1 (mod n) if and only if $ord_{n}b$ | x and then it gives proof for that. Aren't these values already equal? so of course they divide each other or I'm missing something?

Answer 1

There could be lots of exponents that cause $b^x \equiv 1 \pmod{n}$. The order is the least positive of these.

For instance,

$$2^3 \equiv 2^6 \equiv 2^{15} \equiv 1 \pmod{7}$$

and note that $3$ divides each of $6$ and $15.$