Making a $1,0,-1$ linear commbination of primes a multiple of $1000$ 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

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