Or why not also make the period of the moon's orbit harmonic with some other easily observable celestial event? There are innumerable ways we can imagine that aliens could "encode" the artificiality of the moon, and therefore a distinct statistical likelihood that at least one of them will arise by chance.
This is the basis of the fallacy of numerology. When individual events occur with minute probability, it is often wrongly believed that aggregate probabilities of patterns in sequences or sets of those events must necessarily be on the same general order of magnitude as the original events. Statistically that is not the case. In numerology a large number of artificially-created coincidences or significances is silently contemplated, but only a few such "salient" ones are considered. This creates the illusion that it is improbable such a correlation arose by chance.
But since the "salient" correlation (Michael Jackson's shoe size in centimeters, for example, with the synodic orbital period of the moon in days -- the so-called "Moonwalk Ratio" :-) ) is actually drawn from an endless vat of arbitrarily-defined and questionably-relevant correlations, the question at hand asks what the chances are of all those correlations failing. Given enough potential correlations, the chance that at least one of them will arise randomly becomes a fair bet. There are other valid orbital periods besides the synodic. How do we know which was "intended"?
The question is compounded by adding arbitrary tolerances on the metrics. If you say it is a coincidence that the moon's diameter and distance provide for "perfect" eclipses, you have to decide how closely you're going to measure. The eclipses are not perfect, just close enough. If you widen the "window" into which a candidate metric can fit, you greatly improve the chances that it will fit without exhibiting any qualitative correlation.
Saying that the moon's synodic period is 29 days, and Micheal Jackson's shoes are 29 centimeters long seems rigorous. Saying that the moon's synodic period is 29.53 days and Michael Jackson's shoe size is 29.03 centimeters is less impressive. The reader has to be convinced that the "error" is still within a significant tolerance. Unfortunately most readers' intuition about tolerance is more forgiving than statistical tolerance analysis.
And why centimeters? Why days? Why shoes? Why our moon?
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