Is it possible that a - b + 2(c - d) = 5 k

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Tomas314

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Ahoj,
znovu jsem narazil na zajímavý úkol. Dokažte, že je možné zvolit čtyři různá čísla ze sad, která obsahuje šest různých celých čísel, takže [MATH] a_1-a_2 + 2 (a_3-a_4) [/MATH] je násobkem pěti.

Díky.
 
Tomas314 said:
Ahoj,
znovu jsem narazil na zajímavý úkol. Dokažte, že je možné zvolit čtyři různá čísla ze sad, která obsahuje šest různých celých čísel, takže [MATH] a_1-a_2 + 2 (a_3-a_4) [/ MATH] je násobkem pěti. Díky.[/MATH]
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Please use English as the language of communication at this forum. Otherwise your posts will be tagged as SPAM and will be deleted.

Please show us what you have tried and exactly where you are stuck.

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Please share your work/thoughts about this problem.
 
Tomas314 said:
Hi,
I came across an interesting task again. Prove that it is possible to choose four different numbers from a set that contains six different integers, so that [MATH]a_1 − a_2 + 2 (a_3 − a_4)[/MATH] is a multiple of five. Thanks.
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Is Google right?

This seems incomplete. Do we need to know what the set of six integers is, or is this supposed to be true for any such set?
 
Tomas314 said:
Hi, it is supposed to be true for any set in integers.
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Thanks. Now we need to know what you have tried (and what you know that might be useful).

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Without having worked on it in detail yet (and no promise that I will be able to solve it at all), my first thoughts are to work in modulo 5, and to consider the pigeonhole principle. I also find it potentially interesting that you have a linear combination of four terms with every non-zero coefficient mod 5 (1, -1, 2, -2). On the other hand, we don't know whether the 6 integers are different (mod 5).
 
Tomas314 said:
I don't know where to start, this is my problem.
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Well, you start wherever you are! That's why I asked for "what you know that might be useful"! Are you taking a course for which this is an exercise? Then tell us what has been taught that this might be intended to use. Is this a contest problem or a practice for an admission exam? Then tell us what prerequisites they expect of you. The more we know of the context, the more likely we can suggest how to begin using your current knowledge, or what to study.

And in any case, we are here to help you learn to use your own mind, not to give answers, as we've already said (even if I actually had a complete answer to give, which I don't yet).

So, what do you know of number theory and proof techniques? Did any of what I mentioned sound like something you could try?
 
I have a suggestion of a place to start. Consider a simpler version of the original problem...

If you choose 2 numbers from a set of three, consider the possible outcomes from

x=a1-a2 mod 5

Bring all your mod5 answers into the range 0 to 4.

If you can't obtain x=0 then what does this imply about the numbers in the original set of three (mod5)? And, therefore, in this case what values can x be?
 
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