2016 AMC 12A Problem 22

Attempt Problem 22 of the 2016 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2016 AMC 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

22.

How many ordered triples

(x,y,z)

of positive integers satisfy

\text{lcm}(x,y)=72,

\text{lcm}(x,z)=600,

and

\text{lcm}(y,z)=900?

15

16

24

27

64

Answer: A

Concepts:least common multiple prime factorization casework

Difficulty rating: 2160

Small hint:

Factor:

72=2^3\cdot3^2,

600=2^3\cdot3\cdot5^2,

900=2^2\cdot3^2\cdot5^2

Big hint:

Handle each prime

2,3,5

separately, using

\max

of the exponents

Solution:

Because

\text{lcm}(x,y)=2^3\cdot3^2

and

\text{lcm}(x,z)=2^3\cdot3\cdot5^2,

the factor

5^2

divides

z

while neither

x

nor

y

is divisible by

5.

Also

3^2

divides

y,

while neither

x

nor

z

is divisible by

3^2,

and

x

must have the factor

2^3.

Writing

x=2^3\cdot3^{\,j},

y=2^{\,k}\cdot3^2,

and

z=2^{\,m}\cdot3^{\,n}\cdot5^2,

the lcm conditions require

\max(j,n)=1

and

\max(k,m)=2.

The first pair can be

(1,0),(0,1),(1,1),

and the second can be

(2,0),(2,1),(2,2),(0,2),(1,2).

Thus there are

3\cdot5=15

ordered triples.

Thus, the correct answer is A.

Problem 22 in other years