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2020 AMC 10A Problem 9

Problem 9 of 25EasierNumber Theory

A single bench section at a school event can hold either 77 adults or 1111 children. When NN bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of N?N?

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Solution

If the equal number of adults and children is PP, then the adults use P7\frac{P}{7} bench sections and the children use P11\frac{P}{11} bench sections. Thus N=P(17+111)=18P77N=P\left(\dfrac17+\dfrac1{11}\right)=\dfrac{18P}{77}. The least positive integer occurs when P=77P=77, giving N=18N=18. Thus, B is the correct answer.
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Tagged: least common multiple · divisibility

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