Question

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 $n?$

$3$

$6$

$8$

$9$

$12$

Answer: B

Concepts:least common multiple greatest common divisor prime factorization

Difficulty rating: 1140

Solution:

Prime factorize $18=2\cdot3^2,$ $45=3^2\cdot5,$ and $180=2^2\cdot3^2\cdot5.$

The lcm condition forces the exponent of $2$ in $n$ to be $2,$ the exponent of $5$ to be $1,$ the exponent of $3$ to be at most $2,$ and every other prime exponent to be $0.$ The gcd condition $\gcd(n,45)=15=3\cdot5$ further forces the exponent of $3$ in $n$ to be exactly $1.$

Therefore $n=2^2\cdot3\cdot5=60,$ and the sum of its digits is $6.$

Thus, B is the correct answer.

2022 AMC 10A Problem 7

Answer: B

Difficulty rating: 1140

Solution:

Problem 7 in Other Years