Problem Solving

If x represents the sum of all the positive three-digit numbers that can be constructed using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?
(A) 3
(B) 6
(C) 11
(D) 22
(E) 222
[spoiler]Answer
There are 3!, or 6, different three-digit numbers that can be constructed using the digits a, b, and c:

The value of any one of these numbers can be represented using place values. For example, the value of abc is 100a + 10b + c.
Therefore, you can represent the sum of the 6 numbers as:


x is equal to 222(a + b + c). Therefore, x must be divisible by 222.
The correct answer is E.[/spoiler]

We get 6 different numbers

abc = 100a + 10b + c
acb = 100a + 10c + b
bac = 100b + 10a + c
bca = 100b + 10c + a
cab = 100c + 10a + b
cba = 100c + 10b + a

x = Sum of all 6 numbers = 100a + 10b + c + 100a + 10c + b + 100b + 10a + c + 100b + 10c + a + 100c + 10a + b + 100c + 10b + a = 222*(a+b+c).
Since, we don't know the value of a,b and c, 222 is the largest integer by which x MUST be divisible.

p.s: I know that you already provided the solution, but I am on a math-high today!