Divisibility rules for integers are generally known and were summarized, for example, in the first answer to this post:
Divisibility rule for large primes
Using these divisibility rules, one can check if a number is divisible by any prime divisor of $10^k-1$ or $10^k+1$ by checking if the sum or alternating sum, respectively, of its k-digit blocks is divisible by the same factor.
I want to apply the same divisibility rules not to individual numbers but to certain digit patterns. For example, any number that has the digit pattern "ABABAB" would be always divisible by $37$ (which is a divisor of $10^3-1$) because in the corresponding sum of 3-digit blocks ABA + BAB = 111A + 111B both coefficients are divisible by 37.
Not all divisibility rules would be applicable to digit patterns. For example, divisibility by 2 or 5 cannot be checked in this way because the last digit in any pattern is arbitrary, so distinguishing odd and even numbers is impossible at the level of digit patterns.
So far, I identified 3 divisibility rules that apply to digit patterns:
If $p$ divides $10^r-1$, then the divisibility by $p$ can be tested by checking if the sum of $r$-digit blocks is divisible by $p$.
If $p$ divides $10^r+1$, then the divisibility by $p$ can be tested by checking if the alternating sum of $r$-digit blocks is divisible by $p$.
If an integer has 10 distinct digits that all occur an equal number of times, it is divisible by 9 because its sum of digits is a multiple of 45.
I wonder if it is possible to prove that no other divisibility rules exist. In other words, if I find some digit pattern that has a prime divisor that is shared by all integers with that pattern, would it be true that exactly one of the following statements is correct:
(1) this divisor must be one of those prime divisors of $10^r-1$ or $10^r+1$ for which divisibility rules 1 or 2 apply or this pattern has all distinct digits occurring an equal number of times, therefore fulfilling the conditions of rule 3;
(2) the common divisor does not correspond to any of the three divisibility rules, then finding it requires a brute-force calculation of the gcd of all numbers sharing this pattern, and no possibility exists to find a new divisibility rule to simplify this calculation.
To put it short, my question is if one can think of other divisibility rules applicable to digit patterns or prove that none exist. Is this list of divisibility rules complete?
