 Solution

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

Factor

Q = \prod_{i} p_{i}

and

Q^{'} = \prod_{j} q_{j}

, retaining repeated prime factors. If

P

Q Q^{'}

, it has one with at least one of

Q, Q^{'}

, and both sides of the desired equality are zero. Otherwise the definition of the Jacobi symbol gives

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

1

, so the argument includes either denominator being

1

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