Claim: for an integer n greater than or equal to 2, the period of the decimal expression for the rational number 1/n is at most n-1
Proof: I am unsure how to provide a precise proof of this. I am aware that you cannot have a remainder greater than n itself as since there are only n different remainders when you divide by n and once any remainder repeats, the whole process repeats. More precisely tho, we can say that no period will be longer than n-1. I know that the algorithm for finding the decimal expansion of a rational m/n goes:
1) Divide m by n, so we get m=na1+r1 where r1 is the remainder 2) Next divide 10r1 by n, we get 10r1=na2+r2.
If we continue this way, the numbers a1,a2,... are the decimals for m/n. Then remainders are always
however, I think it wants you to show that the period is at most n−1. I am aware that given the algorithm we described above, if there was a period of length n, then the remainders above would have to visit every number between 0 and n-1. But what would happen? How do I write a precise proof