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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*(b-1)=0 mod n, and S must be 0 mod n.
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B) Prove that m = 1 if and only if n is a repunit to the base b, and also that one of the ri is equal to 1.
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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