Factors and Divisors

Thanks to CMA’s ‘Short Circuit’ Volume 13, Number 11, a puzzle/activity that might keep some students entertained on the topic of looking at factors and divisors…

What is a mathematical black hole? It is hard to explain in theoretical terms but the idea is easy to convey using examples. Basically, you choose a number, apply a well-defined mathematical process to that number and then apply the process to the answer. The process is repeated until you get to a point where you can’t change the answer or you get into an endless loop of answers. As in the astronomical properties of a black hole in space, you cannot escape from a mathematical black hole. In the following example the black hole is 15 but it can take a long time to get there.
Take any integer and write down all its divisors including 1 and itself. Add up the digits in the divisors and then repeat the process as many times as you can.
Example: I choose 12 and so the divisors are 1, 2, 3, 4, 6 and 12 with digit sum 19. Then, 19 has divisors 1 and 19 with digit sum 11. This has divisors 1 and 11 with digit sum 3. Then,
1 + 3 = 4
1 + 2 + 4 = 7
1 + 7 = 8
1 + 2 + 4 + 8 = 15
1 + 3 + 5 + 1 + 5 = 15
We are at the black hole of 15 and it took 7 runs to get there.
Does it have to be 15? Are there other possible end points? I haven’t found any – see how you go.