I have started getting pissed at people who snap at someone “Don’t necro this post” (Or any of the numerous other things they say), on information that is well outdated that could fucking seriously use an updated answer.

End rant…I’d prefer not, though…I want to keep this rant going.

In a really twisted way, Youtube is a fairly big reason why a good number of people go outside, take up hobbies, etc.

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I did find that it can be done arbitrarily. Mind is definitely not into writing about it, though, but here’s the gp code I wrote to look it over.

/*
    There may exist a 0<=t<s such that
    s divides both x and (x+(x%d)*(t*d-1))/d.


    To show this for solving for divisibility of 7 in 
    any natural number x.

    g(35,5,10) = 28
    g(28,5,10) = 42
    g(42,5,10) = 14
    g(14,5,10) = 21
    g(21,5,10) =  7
*/

g(x,t,d)=(x+(x%d)*(t*d-1))/d;

/* Find_t( x = Any natural number that is divisible by s,
           s = The divisor the search is being done for,
           d = The modulus restriction ).

    Returns all possible t values.
*/

Find_t(x,s, d) = {
    V=List();
    
    for(t=2,d-1,
        C = factor(g(x,t,d));
        for(i=1,matsize(C)[1],if(C[i,1]==s, listput(V,t))));
        
    return(V);
}   

One thing that I noticed almost right away, regardless what d is, it seems to always work when s is prime, but not when s is composite.

Too tired…Pains too much…Have to stop…But still…interesting.

Yeah, before my mind decided it didn’t like learning any more, I had learned the gist of Bell and Noll’s calc, then switched to gp a few years later…for which I can not remember why, but I can still remember how to use it fairly well.

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, 7*17 = 119 (1*10^2+1*10+9), would be 7*17 = 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).