There is a special place in the netherworld reserved for mathematicians who play with prime numbers. It consists of a corridor 1000 miles long lined with 1,000,000 doors. Each of the 6666 souls imprisoned there is assigned a prime number, 2, 3, 5, 7, 11, etc.
At the start of the first month all of the doors are closed. Each person must walk down down the corridor, opening or shutting every P-th door, according to the given prime. For example, the soul assigned #5 opens or shuts doors 5, 10, 15, 20, etc.
At the end of a month they have all reached the end of the corridor, and they must start back, performing the operation in the opposite order. For example the #7 soul opens or closes doors 999994, 999987, 999980, 999973, and so forth.
How many months will it be before the doors are all closed again?
It's apparent that the counters all reset to zero after completing a run from the fact that #7's first door on the way back is 999994, which is the seventh from the end. There's no leftover amount to be completed from a remainder after the previous run (1,000,000 is not a multiple of 7).
Therefore the same set of doors is affected (toggled) by each pair of run-throughs. So after two pairs of run-throughs, the doors are restored to all closed. The whole process takes 4 months from the beginning of the first phase.
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Posted by Charlie
on 2024-02-11 08:40:31 |
solution
