# Sunday Times Teaser 2664 – Time ‘n’ Again

*by Victor Bryant*

The following sum involves the conversion of a fraction in its simplest form into a recurring decimal:

TIME

——– = .AGAINAGAIN …

**N**

where letters consistently represent different digits and asterisks (*) represent any digit.

Numerically, what is TIME?

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Another version that is much faster along the lines of Arthur’s below:

Similar analysis to Brian, using the factorisation of 99999=3^2*41*271, but using the equation

3^2*41*271 * TIME = AGAIN * xxNxx, and distributing the factors of 99999 amongst AGAIN and xxNxx

And a faster version, using Brian’s observation that **N** has to be 11111 or 33333

Hi Arthur

I hope you don’t mind but I have deleted the incorrect version and, while I was at it, I converted your corrected version to Python 3 (I get emails telling me that some code here doesn’t work and I often find that this is because people are using Python 3 without even realising that there is a Python 2).

We seem to have converged on very similar solutions 🙂