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Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability (resit) / 3 / d / Solution 2026-10-31

The decimal expansion is

a = \sum_{i = 0}^{m} a_{i} 1 0^{i}

. Since

10 \equiv 1 (mod 3)

, repeated use of the multiplication rule in part (b)(ii) gives

1 0^{i} \equiv 1 (mod 3)

for every

i \geq 0

. Repeated use of the addition rule in part (b)(i) now gives

a = \sum_{i = 0}^{m} a_{i} 1 0^{i} \equiv \sum_{i = 0}^{m} a_{i} (mod 3) .

If

3 ∣ a

, the left side is congruent to zero, whence

3 ∣ i = 0 \sum m a_{i} .

This is the decimal digit-sum divisibility test for three, derived directly from modular arithmetic.

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