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

The decimal expansion is $a={\sum }_{i=0}^m{a}_{i}10^i$. Since $10\equiv 1(\mathrm{mod}3)$, repeated use of the multiplication rule in part (b)(ii) gives $10^i\equiv 1(\mathrm{mod}3)$ for every $i\ge 0$. Repeated use of the addition rule in part (b)(i) now givesIf $3\mid a$, the left side is congruent to zero, whenceThis is the decimal digit-sum divisibility test for three, derived directly from modular arithmetic.