If n is a 27-digit positive integer, all of whose digits are

Suppose n = 111,111,111,111,111,111,111,111,111. Then the digits of n sum up to 27 * 1, which is divisible by 9, so n is divisible by both 3 and 9 by the basic divisibility rules given in our Arithmetic book. (Any other 27-digit integer with all identical digits is just a multiple of this first one, so we know that all the other values of n will divide by 3 and 9.) We might conclude that the divisibility rule "generalizes", and that any number whose digits sum to a multiple of 27 divides by 27. Be careful, though! This is exact the sort of natural, intuitive thinking that the testwriters like to exploit, so let's see if we can test a number. Suppose n = 111,111,111,111,111,111,111,111,111. Noticing that 111/3 = 37, we can divide n by 3 and obtain n/3 = 37,037,037,037,037,037,037,037,037. The sum of the digits of this number is 90, so n/3 divides by 9. (n/3)/9 = n/27, so n divides by 27. Since any 27-digit number with one repeating digit is just a multiple of 111,111,111,111,111,111,111,111,111 -- i.e., a multiple of a multiple of 27 -- we conclude that n is divisible by 27.

DO WE HAVE TO WRITE OUT THE WHOLE NUMBER? IS THERE ANOTHER WAY TO SOLVE? ALSO WHY DO WE DIVIDE 111 by 3? AND THEN HOW DO WE GET TO n/3 = 37,037,037,037,037,037,037,037,037?