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My friend asked me this question, and it's been bugging my mind

There are 5€ bills and 8€ bills, what are all the possible bills one can create with these two types of bills. Prove your answer

the only hint is to use strong induction

How can I approach this question, how can I solve it ?

I think it is related with combination, but I couldn't quite justified it in my mind

Thanks

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2 Answers 2

up vote 2 down vote accepted

One can do it crudely. Try to make various quantities. After a not very long time, you will find $5$ consecutive quantities that can be made. After that, anything larger can be made by adding a suitable number of $5$'s. This is because if you have five consecutive numbers $a$, $a+1$, $a+2$, $a+3$, and $a+4$, then any number $b\gt a+4$ differs from one of these by a multiple of $5$.

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Sketch: if you have three 5s, you can replace them with two 8s. If you have three 8s, you can replace then with five 5s. If you have neither three 5s nor three 8s, there's a limit to how much you can have.

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