1987 AIME Problem 7

Question

Let

[r,s]

denote the least common multiple of positive integers

r,

s.

Find the number of ordered triples

(a,b,c)

for which

[a,b]=1000,

[b,c]=2000,

and

[c,a]=2000.

Answer

Treat the exponents of

2

and

5

independently

For each prime, translate every least common multiple into a condition on pairwise maxima

For the exponent of

5,

all three pairwise maxima equal

3.

Thus at least two exponents are

3:

there is one all-

3

triple and

3\cdot3

triples with exactly two

3

’s, for

10

choices.

For the exponent of

2,

the first pair has maximum

3

while the other two have maximum

4.

The exponent of

c

must be

4,

and the exponents of

a

and

b

lie in

\{0,1,2,3\}

with maximum

3,

giving

4^2-3^2=7

choices. Independence gives

10\cdot7=70.

Problem 7 in other years