All eight integers are the same modulo 2005

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Enthurelxyz
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All eight integers are the same modulo 2005

Unread post by Enthurelxyz » Sun Jan 10, 2021 1:12 pm

At each corner of a cube, an integer is written. A $legal$
$transition$ of the cube consists in picking any corner of the cube and adding the value written at that corner to the value written at some adjacent corner. Prove that there is a finite sequence of $legal$ $transitions$ of the given cube such that $8$ integers written are all the same modulo $2005$.
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Mehrab4226
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Re: All eight integers are the same modulo 2005

Unread post by Mehrab4226 » Wed Jan 27, 2021 10:28 pm

All numbers mentioned are in the mod 2005
Now the initial numbering of the cube can be anything. Let there is a vertex with a value of $k$, and also has an adjacent vertex with the $m \neq 401,5$ then we can use $k$ as the point use a legal transition. This transition will change the number $k$ to something else. In repeated use of the same transition, we can have $2005$ different values on the vertex having $k$ initially. Why? Because all are in mod $2005$ if a transition brings a number that already came before, it means we added a multiple of $2005$ with $k$. That if $k+m \times g \equiv k $ then $m \times g$ is a multiple of $2005$. that is only possible if $g = 2005$ since m is not a factor of $2005$. So we can have $2005$ different values on the vertex upon transition. So one of those values must be $0$. We will continue it until we have seven $0$ vertices. Finally, we have a non-zero vertex which we can use to make the other vertices the same value as it(non-zero vertex) and fulfilling the property of the question. Now if we have $401$ or $5$ among the vertices we can use a transition of $401,5$ with any number from $(1-2004)$ and get a number, which is not a factor of $2005$. Which we can use as a replacement of $m$ and do the same thing over again. Thus we can use a finite number of transitions to make all of the vertex equal in mod $2005$. $\square$
This was a constructive proof. I have skipped a few details. If you do not understand a portion feel free to notify me. :)
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

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