A convex figure P is called a reptile if it can be decomposed into \(k\ge 2\) nonoverlapping and congruent figures similar to P. It is known that every convex reptile is either a triangle or a trapezoid. The characterization of reptile trapezoids is open; it is not known, e.g. if there are infinitely many nonsimilar reptile trapezoids other than the parallelograms. We present some number-theoretical necessary conditions that reptile trapezoids must satisfy. We show that if a reptile trapezoid is not a parallelogram and not a right trapezoid of particular types, then its sides are pairwise commensurable. We present several consequences of this fact, including the countability of nonsimilar reptile trapezoids, not a parallelogram.
