Arithmetic progressions with coprime differences Suppose we have finite number $n \geqslant 2$ of arithmetic progressions $\{x \equiv  r_1 \pmod {d_1}\}, \ldots ,\{ x \equiv r_n \pmod {d_n}\}$ such that $\gcd(d_1, \ldots, d_n) = 1.$
Is true that some pair of them has nonempty intersection? (I think, it's true).

Counterexample: the three congruences $x\equiv 0\pmod{6}$, $x\equiv 1\pmod{10}$, $x\equiv 2\pmod{15}$.
One can produce similar counterexamples using any three distinct primes $p,q,r$ instead of $2,3,5$. 
Remark: I notice that fedja gave a very similar counterexample in a comment. 
