 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2021/1/a/iii/solution

p

(P P^{'} / p) = (P / p) (P^{'} / p)

, including the case of a zero factor. Therefore, writing

Q = \prod_{i} p_{i}

with repetitions,

(\frac{P P ^{'}}{Q}) = i \prod (\frac{P P ^{'}}{p _{i}}) = i \prod (\frac{P}{p _{i}}) i \prod (\frac{P ^{'}}{p _{i}}) = (\frac{P}{Q}) (\frac{P ^{'}}{Q}) .

This product definition still agrees with the stipulated zero when the numerator and denominator are not coprime. For

Q = 1

all three symbols are

1

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