Adding 2 character puzzle

Problem:

ABC + DEF + GHI = JJJ

If different letters represent different digits, and there are no leading zeros, what does J represent?

Solution: 

The value of J must be 9.
Since there are no leading zeros, J must be 7, 8, or 9.
(JJJ = ABC + DEF + GHI = 14? + 25? + 36? = 7??)

Now, the remainder left after dividing any number by 9 is the same as the remainder left after dividing the sum of the digits of that number by 9. Also, note that 0 + 1 + ... + 9 has a remainder of 0 after dividing by 9 and JJJ has a remainder of 0, 3, or 6. The number 9 is the only number from 7, 8 and 9 that leaves a remainder of 0, 3, or 6 if you remove it from the sum 0 + 1 + ... + 9. Hence, it follows that J must be 9.