2022 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 7 Easy

The least common multiple of a positive integer n and 18 is 180, and the greatest common divisor of n and 45 is 15. What is the sum of the digits of n?

  • A.

    3

  • B.

    6

  • C.

    8

  • D.

    9

  • E.

    12

Answer:B

\because lcm(n,18)=180

\therefore 2^2 \cdot 5 | n and n | 2^2 \cdot 5 \cdot 3^2=180,

\therefore n=2^2 \cdot 5 \cdot m (m is a factor of 3^2).

\because gcd(n,45)=15,

\therefore 3 \cdot 5 | n and 3^2 \cdot 5 | n,

\therefore If you factorize n, there is only one factor 3,

\therefore n=2^2 \cdot 5 \cdot 3=60.

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