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Source. Proposition 2, PDF p. 2 of arXiv:math/0604347v2.

Statement

For two congruence classes $a_i\pmod{m_i}$ and $a_j\pmod{m_j}$, no integer lies in both exactly when

$$\gcd(m_i,m_j)\nmid a_i-a_j.$$

Proof scope. The paper introduces the criterion as the one underlying its pigeonhole argument for $k=3$ on pp. 1--2. No separate proof accompanies the numbered statement, and no independent proof review is claimed here.

Bears on. Problem 202: the criterion decides pairwise disjointness, the condition on the families that problem counts. The paper does not apply it to that problem.