2016 AMC 10A Problem 25

Question

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

Treat each prime exponent independently

The powers of

3

and

2

each leave a small max-condition count

Factor the given least common multiples:

\begin{gathered} 72=2^3\cdot3^2, \\ \quad 600=2^3\cdot3\cdot5^2, \\ \quad 900=2^2\cdot3^2\cdot5^2. \end{gathered}

Since

\operatorname{lcm}(x,y)

has no factor of

5

, neither

x

nor

y

has a factor of

5

, and

z

must contain

5^2

.

Write

x=2^a3^b

,

y=2^c3^d

, and

z=2^e3^f5^2

. The power of

3

in

\operatorname{lcm}(y,z)

forces

d=2

, and the power of

2

in

\operatorname{lcm}(x,z)

forces

a=3

.

The remaining independent conditions are

\max(b,f)=1

and

\max(c,e)=2

. These have

3

and

5

ordered choices, respectively, so there are

3\cdot5=15

triples.

Thus, the correct answer is A.

Problem 25 in other years