Suppose that m = 1 mod b. What integer between 1 and m-1 is equal to b^(-1) mod m?
The Attempt at a Solution
m = 1 mod b means that:
m = kb + 1 for some integer k
Let x be the inverse of b mod m, note: x exists since b and m must be coprime due to the previous statement.
xb = 1 mod m
thus: xb = gm + 1 for some integer g.
Now this is were I have little success. I cant seem to manipulate anything to my advantage and I'm unsure how to proceed.
I did find x = (m+1)/b but that is not always an integer. Thanks for any help you can provide.