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?

2022 AMC 12A Problem 4

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The least common multiple of a positive integer

n

and

18

180,

and the greatest common divisor of

n

and

45

15.

What is the sum of the digits of

n?

3

6

8

9

12

Factor

180=2^2\cdot3^2\cdot5

and see which powers

n

must supply beyond

18=2\cdot3^2

\gcd(n,45)=15

forces exactly one factor of

3

and at least one factor of

5

Since

180=2^2\cdot3^2\cdot5

and

18=2\cdot3^2,

the condition

\operatorname{lcm}(n,18)=180

forces

n

to contribute

2^2

and

5,

with its power of

3

at most

2.

From

\gcd(n,45)=\gcd(n,3^2\cdot5)

=15=3\cdot5,

the power of

3

n

is exactly

1

and the power of

5

is at least

1.

Therefore

n=2^2\cdot3\cdot5=60,

whose digits sum to

6.

Thus, the correct answer is B.