A woman named Alice, a distingushed astronomer, along with a group of her students, discovered a planet with a number of moons orbiting it. Alice decided to ignore the planet and concentrate on analyzing these moons.

She found that she could divide the moons equally among herself and her students, so that each person would have the same number of moons to study.

But her most brilliant student, Phil, suddenly suggested that the moons be broken up by families rather than individuals, so that people of the same family would work on a group of moons together. The only family relationships among these people were 3 distinct pairs of sisters. When the moons were divided along family lines, it was again possible to divide them so that each family group (each family group consisting of at least one person) got the same number of moons.

Before any study could commence, however, Pierre, another of Alice's students, decided not to study these moons, but to concentrate on the planet itself. With Pierre's decision, Alice at once dropped the division by family scheme, and divided the moons evenly among all the remaining people, including herself.

Two questions: What is the fewest number of students Alice could have had, and the fewest number of moons, to satisfy the requirements of this story ?

I think the answer is 8 students, and 72 moons.

Unless there are female students named Pierre and Phil in her class (technically you don't know because no s/he pronouns were used) you know that she has at least 8 students.  This is because of the 3 distinct pairs of sisters (so there's 6) and Pierre and Phil make 8.

So now the different ways Alice divided her class was between 9 people (all 8 students plus herself), between 6 people (3 pairs of sisters plus the other 2 students and herself), and between 8 people (only 7 of her students, plus herself).

The least common multiple of 9, 6, and 8 is 3^2*2^3 = 72.  So that's how many moons there need to be.

If for some reason Phil and Pierre are girls, I guess the answer is 6 students, but then you'd need 84 moons.  (LCM of 7, 4, 6).

Later!

Posted by nikki on 2004-03-30 10:19:12