Because the remainders when 10, 10^2, 10^3, etc. are divided by 3 or 9 are all 1, the "remainders" when 10c, 10^2b, 10^3a, etc. are divided by 3 or 9 should be c, b, a, etc., respectively. Therefore, to determine whether a natural number is divisible by 3 or 9, we need to check whether the sum of the decimal digits of the number is divisible by 3 or 9.
Now let's look at the case of quickly determining whether the divisor is 11. Since the remainders when 10, 10^2, 10^3, etc. are divided by 11 are -1, 1, 1, etc., it can be easily seen that if the sum of the digits in the units, hundreds, thousands, etc. of a number, minus the sum of the digits in the tens, thousands, hundreds of thousands, etc. of the number, is a multiple of 11, then we can conclude that the number is divisible by 11.