There is some natural number written on the board. An operation is defined as adding to the number its maximal proper divisor. Prove that after some iterations of this operation that we get a number that's divisible by 32020.
(In reply to
a pattern by Charlie)
I think the time it takes for the pattern to emerge is related to how many factors of 2 are in the first number you reach which is 0 mod 4.
So to rearrange/restate my earlier thoughts:
If you start with an odd number, in one step it will be even (either 0 mod 4 or 2 mod 4).
If you have a number that is 2 mod 4, after two operations it will be 0 mod 4.
Once you get to a number that is 0 mod 4, you basically keep multiplying by 3/2 until it's odd - that's why the more factors of 2 it has, the longer it takes. Once you reach the point that it's odd, it begins the cycle of tripling every three operations (because the next operation will multiply it by 4/3, and now since it has 2 new factors of 2, you multiply by 3/2 twice in a row - so three successive operations apply a multiple of 4/3 * 3/2 * 3/2 = 3).
Edited on March 18, 2021, 2:54 pm
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Posted by tomarken
on 2021-03-18 14:51:04 |
re: a pattern
