divisble by 3

Question

an integer n is divisible by 3 if and only if the sum of digits of n is divisible by 3. In other words, if n= a0+ 10 a1 +102 a2 +…+ 10k ak, then n is divisible by 3 if and only if a0+a1+a2+…+ak is divisible by 3. Example: 12765312 is divisible by 3 because 1+2+7+6+5+3+1+2=27 is divisible by 3; indeed, 12765312/3 = 4255104.

Answer 1

Take a 3-digit number as an example: if 100a+10b+c=3n where n is an integer, then we can write this as:

(99a+9b)+(a+b+c)=9(11a+b)+(a+b+c)=3n, therefore the sum of the digits:

a+b+c=3n-9(11a+b)=3(n-3(11a+b))=3p where p=n-3(11a+b) is an integer. This means that the sum of the digits is also divisible by 3.

10^x-1 is always divisible by 9, hence it is also divisible by 3: 10^x-1=9m where m is an integer.