Time Bomb

\(n\) people stand in a circle. One is given a ticking bomb with \(n-1\) seconds left. The bomb is then thrown around the circle every second in an orderly, mathematical fashion; when the bomb has \(k\) seconds left, it is thrown to the person \(k\) places clockwise. Determine the sum of all values of \(n\) with \(1 \le n \le 100 \) for which each person holds the bomb exactly once.

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