Problem:
How many ordered triples (x,y,z) of positive integers satisfy lcm(x,y)=72, lcm(x,z)=600, and lcm(y,z)=900?
Answer Choices:
A. 15
B. 16
C. 24
D. 27
E. 64
Solution:
Because lcm(x,y)=23⋅32 and lcm(x,z)=23⋅3⋅52, it follows that 52 divides z, but neither x nor y is divisible by 5 . Furthermore, y is divisible by 32, and neither x nor z is divisible by 32, but at least one of x or z is divisible by 3 . Finally, because lcm(y,z)=22⋅32⋅52, at least one of y or z is divisible by 22, but neither is divisible by 23. However, x must be divisible by 23. Thus x=23⋅3j,y=2k⋅32, and z=2m⋅3n⋅52, where max(j,n)=1 and max(k,m)=2. There are 3 choices for (j,n) and 5 choices for (k,m), so there are 15 possible ordered triples (x,y,z).
The problems on this page are the property of the MAA's American Mathematics Competitions