A number is divisible by three iff the sum of its digits is divisible by three. First, prove 10^N = 1 mod 3 for all integers N >= 0. 1 = 1 mod 3. 10 = 1 mod 3. 10^N = 10^(N-1) * 10 = 10^(N-1) mod 3. QED by induction. Now let D0? be the units digit of N, D1? the tens digit, etc. Now N = Summation From k=0 to k=inf of Dk?*10^k. Therefore N mod 3 = Summation from k=0 to k=inf of Dk? mod 3. QED

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