I had a thought experiment that I was interested in formalising. I don’t have a maths background so apologies.

**How can you calculate the possible population sizes for a given average of whole numbers?**

Assuming that the number of significant digits is infinite, the population consists of whole numbers only and, for example, the average is exactly 0.9.

What I have worked out (and think is correct) is that for any population size, N, the decimal part of the output will be caused by a remainder going from 1/N to N-1/N. So the denominator of any fraction that reduces to 0.9 is in the set of possible populations and the minimum population size is therefore 10 i.e 9/10 going up to infinity

My question is how is this more formally worked out. Is there a name for all of the fractions that reduce to 0.9 or are they simply multiples of 0.9?

Are there any other tips or tricks I missed?