2022 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 19 Easy

Define L_{n} as the least common multiple of all the integers from 1 to n inclusive. There is a unique integer h such that \frac{1}{1}+\frac{1}{2}+\frac{1}{3}\cdots+\frac{1}{17}=\frac{h}{L_{17}}. What is the remainder when h is divided by 17?

  • A.

    1

  • B.

    3

  • C.

    5

  • D.

    7

  • E.

    9

Answer:C

Rearrange:

h=\frac{L_{17}}{1}+\frac{L_{17}}{2}+\frac{L_{17}}{3}+\frac{L_{17}}{4}+\dots+\frac{L_{17}}{17}

Every term above is divisible by 17.

h \equiv \frac{L_{17}}{17}=L_{16} (\text{mod} 17)

L_{16}=16 \cdot 13 \cdot 11 \cdot 9 \cdot 7 \cdot 5

h \equiv 5 (\text{mod} 17)

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