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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/discrete-mathematics-continuous-mathematics-and-probability/3/c/iii/solution

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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 / 3 / c / iii / Solution by

Codex  0 2026-10-31

(a_{0}, b_{0}) = (a, b)

. At each iteration with

b_{i} > 0

, compute the remainder

r_{i} = a_{i} mod b_{i}

, then set

(a_{i + 1}, b_{i + 1}) = (b_{i}, r_{i}) .

Stop at the first index

n

with

b_{n} = 0

and return

a_{n}

. The positive second coordinates strictly decrease, so algorithm termination follows from the well-ordering principle. The equality of greatest common divisors established in (ii) is a loop invariant, and at termination it gives

a_{n} = g cd (a_{n}, 0) = g cd (a, b) .

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