Mathematical Swimmer

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Every day I go to the swimming pool and swim the same number of lengths. I like to count the number of lengths I've done as I go as a fraction of the total number of lengths I'm going to do that day.

If I swam ten lengths a day, after five lengths I would say to myself, "I've managed $\frac{5}{10}$ of my day's swimming - that's $\frac{1}{2}$!" After eight lengths, I would say I'd done $\frac{8}{10}$, which simplifies down to $\frac{4}{5}$. After nine lengths, I'd say I'd done $\frac{9}{10}$, which does not simplify.

swimmer

I don't swim ten lengths a day. In fact, the total number of lengths I swim each day is rather special.

Each number of lengths I swim will either be a prime number or the fraction it makes of the total number of lengths will simplify. 
It is, in fact, the largest number possible for which this is true.

How many lengths do I swim per day?

Published May 2000.