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This is a classic problem from Discrete Mathematics. Consider the following sums: a1, a1 + a2, a1 + a2 + a3, ..., a1 + a2 + ... + an. By pigeonhole principle, either one of these sums is exactly 0 mod n or two of these sums have the same remainder mod n. Hence, their difference will be 0 mod n.
- saumya January 26, 2015This can be translated into a linear time algorithm which computes the sums: a1, a1 + a2, a1 + a2 + a3, ..., a1 + a2 + ...+ an. Also, at the same time computes their remainder modulo n. Then, amongst the remainders we search for 0 or two identical remainders. All of these can be done in linear time