Competition problem — Number theory

Library / /109 of 520

Number theory Difficulty 5.2 AIME, harder Find the answer

10. To pack books when moving a school library, you can buy small boxes that hold 12 books or large ones that hold 25 books. If all the books are packed in small boxes, 7 books will remain, and if all the books are packed in large boxes, there will be room for 5 more books. The library's collection contains between 500 and 650 books. How many books are in the library?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Solution: $\left\{\begin{array}{ll}a=12 R+7=12(R+1)-5, & a+5 \vdots: 12, \\ a=25 l-5, & a+5 \vdots 25,\end{array}\right\} a+5: 300$ . Answer: $a=595$ .

$a+5=300, \underline{600,} 900$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.