There are some books on the table. If you group them by 3, you get some number of full groups and 2 books remain; if you group them by 4, you get some number of full groups and 3 books remain; if you group them by 5, you get some number of full groups and 4 books remain. What is the number of books on the table, if it is less than 100?
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What would the situation be if there were one more book? – Joffan Sep 04 '17 at 04:38
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Consider adding $1$ more book to the table.
Then, when you group them in $3$s, then you can do so evenly with no books left over (since the extra book completes the $2$ books left over from before).
Likewise, you can now also group them evenly by $4$s and by $5$s with no books left over.
Thus, the new number of books is divisible by $3$, $4$, and $5$, so it is a multiple of $\operatorname{lcm}{\left(3,4,5\right)}=60$. Since it must be less than $100$, it must be $60$.
Subtracting the extra book from the start leaves $\boxed{59}$ books.
Ant
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