I agree with you that “even” comes up much more often in typical mathematical and non-mathematical discourse than divisibility by anything else. That having been said, “exactly divisible by three”, “divisible by three with remainder of 1”, and “divisible by three with remainder of 2” give just as good a system of arithmetic as the one you are using with even and odd.
Yes, of course the concepts of even and odd far, far, far predate the base-10 notational system; most simple arithmetic concepts do. The reason we have a word for divisibility by 2 isn’t because 2 happens to divide into 10. We have the word because we’ve been talking about lining things up into pairs for a long time, even long before we had anything like the general concept of division or divisibility (which in turn, as pointed out, came far before numeric notation had standardized to anything like the current system).
But “modulo 3” isn’t “a word”, so I’m curious why you think that is better than “divisible by 3”, especially since the latter would be more widely understood than the former.
Hypotenusal. Running around a triangle, every third leg is the hypotenuse, and would be number of leg passages divisible by three. Since only triangles have hypoteni, this would be unambiguously every third leg.
Which is a consequence of using base 10 and 10 being congruent to 1 base 3. It works for divisibility by 9 too. Odd/evenness extends for powers of 2 in that divisibility by 2^N requires that you simply check the last N digits - divisible by 4 if the last two digits are, etc. Same for 5 and powers thereof. Consequence of them factoring the base. You may have been taught quick divisibility checks for 2,3,4,5,6,8,9 back in grade school without being taught why they worked.
I think the only word meeting the OP’s criterion is “triable”, which the Oxford English Dictionary defines as “divisible into three”. All the other candidates presented in this thread (trisectable, ternary, tripartite, trivisible, trifold, triune) plus a few others not yet mentioned (trichotomous, trichotomic, etc.) don’t quite fit or aren’t attested. “Ternary” is a near miss; the OED says that this can mean “a number which is a multiple of three”, which makes it a noun, not an adjective.
Note that these senses of “ternary” and “triable” are marked as rare and obsolete, respectively. I wouldn’t use them in everyday conversation or writing without explicitly defining them first.
I’ve been pondering creating a suitable word based on “aliquot”. “Contained in the whole an integral number of times.” One of Mrs. FtG’s favorite tech words she uses at home.
“Trialiquot” is a mouthful and easily mispronounced. “Triliquot” may not be a kosher formation.
The origins are Latin “alius” (other/another) and “quot” (as many). The latter seems more significant at the former, so maybe “triquot” (pronounced like “try-qwat”).
Then you could have “quadquot”, “quintquot”, etc.
But these would all mean “put into X parts”. Divisibility in thirds might be “triquotable”. Ugh, that’s bad.