Two corollaries of ‘Divisibility of 11'

We notice that in decimal (10-base system), we can evaluate the divisibility of 9 by adding up all its digits and test the divisibility of the sum of the digits. While also in decimal system, we can evaluate the divisibility of 11 by adding up its odd digits and even digits and test the divisibility of the differences of the sum of the digits.

Following the logic in the article that discussing the divisibility of 9, we can made the following corollary: Similar to the idea that we can test test the divisibility of the differences of the sum of the digits, we could also test the divisibility of 12 by the differences of the sum of the odd and even digits in a base 11 system, the divisibility of 13 by the differences of the sum of the odd and even digits in a base 12 system…. etc.

Moreover, I made the hypothesis that in decimal system, divisibility by 12 could be tested by the divisibility of the weighted differences of the sum of the digits, i.e. even digits sum*2- odd digits sum. Moreover, divisibility by 13 could be tested by the divisibility of the weighted differences of the sum of the digits, i.e. even digits sum*3- odd digits sum, divisibility by 14 could be tested by the divisibility of the weighted differences of the sum of the digits, i.e. even digits sum*4- odd digits sum…. etc.
And in general, divisibility by n+x in a n-base system could be tested by the divisibility of the weighted differences of the sum of the digits, i.e. even digits sum*n- odd digits sum(in the same system.)