Question 14
If $$m$$ and $$n$$ are positive integers such that $$30mn - 6m - 5n = 2019$$, what is the value of $$30mn - 5m - 6n$$?
Solution
Adding 1 to both sides factorises the left side as $$(6m-1)(5n-1) = 2020$$. Since $$6m-1$$ must leave remainder 5 on division by 6, the divisors of $$2020 = 2^2 \times 5 \times 101$$ allow only $$6m-1 = 5$$ with $$5n-1 = 404$$, giving $$m = 1$$ and $$n = 81$$. Then $$30mn - 5m - 6n = 2430 - 5 - 486 = 1939$$.
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