Periods in digits and remainders

Periods in digits and remainders

I'm preparing a post with a solution for the first problem in the post . And there is a small question which has irked me for a while.

When we calculate digits of 1/n, there is a function which gives a digit from the remainder. So a cycle in the remainders leads to a period in digits, therefore period of digits can't be longer than period of the remainders. But why can't it be shorter?

I think that I have a glimpse of the idea, how to explain this. But if you have a good explanation, please share it.