Final answer:
The order of an element and its inverse in modular arithmetic is the same, because the inverse element raised to the order of the original element will also be congruent to 1 modulo m, thus having the same order.
Step-by-step explanation:
To demonstrate that if b is an inverse of a modulo m, then the order of b mod m is the same as the order of a mod m, let's consider the definition of the order of an element in a group. The order of an element a in the modulo m group is the smallest positive integer n such that an ≡ 1 (mod m). If b is the inverse of a, we have ab ≡ 1 (mod m). This means that b is also an element of the modulo m group whose order must be found.
If the order of a mod m is n, it means that an is the lowest power of a that is congruent to 1 modulo m. Since b is the inverse, it satisfies the equation bn ≡ a-n ≡ (an)-1 ≡ 1 (mod m). This demonstrates that the lowest power of b that is congruent to 1 modulo m is also n, which means that the order of b mod m is the same as the order of a mod m.
It is worth noting that this argument relies on the fact that the modulo m group is a finite group and that the orders of elements within this group are well-defined. By this reasoning, we can conclude that the order of an element and its inverse in the context of modular arithmetic are the same.