 Solution

ID: 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

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) .

3 ∣ a

, the left side is congruent to zero, whence

3 ∣ i = 0 \sum m a_{i} .

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