Quote:
Originally Posted by Dr Sardonicus
A) Prove that for a positive integer m.

If you multiply the elements of the set {b^i for all i} by b, you get the same set with permuted elements. Thus multiplying the sum by b does not change it mod n. Thus S*b = S mod n, S*(b1)=0 mod n, and S must be 0 mod n.
Quote:
B) Prove that m = 1 if and only if n is a repunit to the base b, and also that one of the r_{i} is equal to 1.

The "and also that one of the r
_{i} is equal to 1" part seems ambiguous or wrong. For k != 1, m may or may not equal 1?
n = 1111, b = 10, k = 2: m = 1
n = 1111, b = 10, k = 21: m = 2