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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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