Not sure, (“Older and a lot more decrepit” doesn’t mean “younger an a lot more mentally sound”, heh. Do wish I could change that, but meh, I can’t).
Anyway, I did find a method similar to what you wrote, so I can redefine it in your terms.
A base 20 number is divisible by 7 if the difference between 8 times the last digit and the remaining digits is divisible by 7.
Ok, a little description on a base 20 number (Think Mayan and Nahuatl/Aztec numbers). 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19 should be considered single digits. So a base 10 number, 717 = 119 (110^2+110+9), would be 717 = 5:19 in a base 20 system (5*20+19).
is 1:8 divisible by 7? (28 in base 10). 8*8 = 3:4. 3:4-1 = 3:3
is 3:3 divisible by 7? 8 * 3 = 1:4. 1:4 - 3 = 1:1 (1*20+1 = 21).
is 9:2 divisible by 7? (182). 2*8 = 16. 16-9 = 7 Check.
I’ll just leave that there. So a long weird way of saying, yes, that’s pretty much my reasoning, but not exactly at the same time. As the first message included the base 20 numbers divisible by the base 20 single digits 7, 13, and 17. (Hopefully that came off a little better).
(Note: Saying “base 20 number[s]” is not important overall. Just being overly descriptive to differentiate between base 10 digits and base 20 digits).