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This is not a homework problem, but the one I've run into in real life. I have a subway card on which the remaining balance is \$2.28. One ride on the subway costs \$2.75, and when you refill the card with money, the subway company gives you an 11% bonus (so when you put in \$10, you actually get \$11.10).

I would like to find the least amount (in dollars and cents, so no sub-cent solutions) I can add to get this card to get an exact number of rides, i.e. a multiple of \$2.75.

It sounds like we have to solve $$2.28 + 1.11k = 2.75N$$ with $k,N$ whole and $k$ minimal positive. This is equivalent to $$228 + 111k = 275N$$ and you can factor the left-hand side $$3 \cdot(76 + 37k) = 275 \cdot (3n)$$ where $N=3n$ and $n$ is a positive integer.

Factoring this into primes we get $$3 \cdot \left(2^2 \cdot 19 + 37k\right) = 5^2 \cdot 11 \cdot (3n)$$ Is there a way to solve this completely without trial-and-error by computer? How would I proceed?

Thank you very much.

UPDATE From numerical simulation, I see that the smallest possible such $k$ is 102 and we end up with $$228 + 111 \cdot 102 = 11,550 = 275 \cdot 42.$$ But how would I come up with this analytically?

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    could the downvoter please explain the downvote? – 2017-07-03

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You have $111k = 275N - 228$. For $k$ to be an integer, you would need $$ 275N \equiv 228\pmod{111}$$

You can find many such $N$ by inverting $275\equiv 53\pmod{111}$.

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    So you suggest I numerically try every one of $N$ and checking if it satisfies $53N \pmod{111} \equiv 0$? – 2017-01-17
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    Suppose you have a number $m$ such that $53m\equiv1\pmod{111}$. Then multiplying both sides of the equation above by $m$ gives $N\equiv 228m\pmod{111}$. – 2017-01-17
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    You can find $m$ without trial-and-error by factoring $111=3\times37$, and then solving for those two primes, combining the results via the Chinese Remainder theorem. – 2017-01-17
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    In particular, $m=44$, so $N\equiv44\times228\equiv42\pmod{111}$, so taking $N=42$ gives the smallest $k$, 102. – 2017-01-17