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## 'Mathematical Swimmer' printed from http://nrich.maths.org/

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.

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?