Basketball t-shirts

[Guest]

The sponsor of a basketball team has offered 209 branded t-shirts to the regional high school players. Each of the t-shirts had a unique number from 2 to 210 printed on its back.
In order to randomly assign the t-shirts to the students, the kids were all given a unique number from 1 to 209 and the shirts would be randomly assigned by carrying out some kind of draw.
The maths teacher, however, decided to make it a bit more tricky, so he asked the students to find a method to assign the shirts in such a way, that each student's number is a factor of his t-shirt's number. He then asked them to find how many such ways there are and also to give an example of such an arrangement (we remind that each number has at least two factors, 1 and the number itself).

[Guest]

One such arrangement is everyone gets the shirt with their number on it, except for student 1 who gets shirt 210.
There are 75 arrangements that would work.

The trick is to decide who gets shirt 210, say it's student 105, then decide who gets shirt 105, say it's student 5, which means shirt 5 must go to student 1.
Clearly a chain is made by removing factors 210 --> 105 --> 5 --> 1.
The example I gave at the start would have a chain 210 --> 1.
There are 75 such chains.

Once you have decided on the chain, the other shirts must go to the students with the same number. This is because the set of students not assigned a shirt, and the shirt numbers not assigned are precisely the same, so the student with the largest number not yet assigned a shirt (student 209) must get a shirt number which is at least as big as their number (shirt 209, or shirt 209x2=418, or 209x3=627, etc, but 418, 627, etc aren't valid shirt numbers), there is only one shirt not yet assigned that satisfies this property and that is the shirt which matches the students number (shirt 209). So we must assign the largest shirt number to the student with the largest id. After we do this we are back in the situation of having the set of ids of students without shirts, being the same as the set of shirt numbers not assigned. So we must repeat the process of assigning the largest shirt (shirt 208) to the student with the largest id (student 208), etc. until we run out of shirts/students.

Hope this helped,

R. Baber.

[Guest]

Just a quick update. The number 75 is from the Fubini series, see
https://oeis.org/A000670
Half way down the comments section they say:

A slightly more transparent interpretation of a(n) is as the number of 'factor sequences' of N for the case in which N is a product of n distinct primes.

In your case N=270, n=4, and a(n) = a(4) = 75.

Hope this helped,

R. Baber.

[Guest]

Sorry meant to say N=210 (not N=270).

R. Baber.

[Guest]

Perfect solution. Thanks!!