Fermat discovered
an amazing fact about prime number clocks [click
here to find out what a clock calculator is]:
Take any hour on the clock with p hours on, multiply
it together p times and as if by magic you return
to the time you started with.
Try the calculation on some prime and non-prime
clocks. For example:
| Powers of 2 |
21 |
22 |
23 |
24 |
25 |
26 |
27 |
28 |
29 |
210 |
| On a conventional calculator |
2 |
4 |
8 |
16 |
32 |
64 |
128 |
256 |
512 |
1024 |
| On a 5-hour clock calculator |
2 |
4 |
3 |
1 |
2 |
4 |
3 |
1 |
2 |
4 |
| On a 6-hour clock calculator |
2 |
4 |
2 |
4 |
2 |
4 |
2 |
4 |
2 |
4 |
So 25=2 (modulo 5) since 5 is a prime,
but 26 is not 2 (modulo 6).
The mathematician Euler
eventually came up with a proof for why this magic
would always happen on prime number clocks. [click
here to see Euler’s proof]
Notice that as the clock hand maps out the hours,
a pattern emerges. After (p-1) steps we are guaranteed
at the next step to return to the time we started.
So the pattern repeats itself every (p-1) steps.
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