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Prove that with every given 10 primes $p_1,p_ 2,\ldots,p_{10}$,there always exist 10 number which are not simultaneously equal to $0$, get one of three values: $-1$, $0$, $1$ satisfied that: $\sum\limits_{ i=1}^{10}a_ip_i$ is a multiple of 1000

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    Please post one question at a time, or explain the relationship you perceive between the questions.2012-12-10
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    thanks for your comment. i have just edited:). To Gerry Myerson: that is symbol familiar in my country, so sorry, edited of course2012-12-10

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The pigeonhole principle takes care of this. There are $2^{10}-1=1023$ non-empty sums, so two are congruent modulo $1000$, etc., etc.

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    please more specific, if there are two sum congruent modulo 1000, what do you do next? for example, if $p_1$ exists in two sum, how can you contruct a sum satisfies that $a_1=1$2012-12-10
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    I'm intentionally leaving some work for you to do. Don't you want the pleasure of solving a problem on your own?2012-12-10
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    ok.i proved. thanks.2012-12-10
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    Good. Now *you* can post a complete solution here. After a while, you can even accept your own solution.2012-12-10
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Following the guide of Gerry Myerson, i think here is solution of this problem. First, we'll built 1,0 linear combination of 10 primes. There are $2^{10}$ combination except for one Zero combination, so we get $2^{10}-1=1023$ non zero combination. Then exists two of them congruents modulo 1000, take their subtraction, we have 1,0,-1 linear combination which is multiple of 1000. This is not so difficult problem, and we can replace "prime" hypothesis by "abitrary integer". and the number 1000 can replaced by $1

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    Out of curiosity: What is the linear combination in this case?2012-12-10
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    @A.Schulz, the linear combination (or combinations --- there could be several) will depend, of course, on which $10$ primes are given.2012-12-10
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    @GerryMyerson: Oh I see. I misread the question. I thought the $p_i$s are the first 10 primes.2012-12-11
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    @A.Schulz, for the first $10$ primes, you can take $2+3-5$, with all the other coefficients zero, or $3+5+11-19$, or any of a number of other solutions.2012-12-11
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    @Schulz:don't let your intention to hypothesis "prime". this solution can apply for any 10 integer^^2012-12-11