Any set of m positive integers contains a nonempty subset whose sum is a
multiple of m.
Proof. Suppose a set T has no nonempty subset with sum divisible by m. Look at the
possible sums mod m of nonempty subsets of T. Adding a new element a to T will give at
least one new sum mod m, namely the least multiple of a which does not already occur.
Therefore the set T has at least |T| distinct sums mod m of nonempty subsets and |T| < m.
Please help me understand the proof. I know modular number theory etc, but I do not understand the logics here.