C3. The addition sum $X C V+X X V=C X X$ is true in Roman numerals.
In this question, however, the sum is actually the lettersum shown alongside, in which: each letter stands for one of the digits 0 to 9 , and stands for the same digit each time it occurs; different letters stand for different digits; $X C V$ and no number starts with a zero.
$$
+\frac{X X V}{C X X}
$$

Find all solutions, and explain how you can be sure you have found every solution.
Solution
First, since $C$ is the leading digit of a three-digit number, $C>0$.
From the tens column we see that $C+X$ results in an $X$. Since $C$ cannot be 0 , there must be a carry digit of 1 from the units column, and $C+1=10$. Therefore $C=9$ and $V$ is at least 5 .
Now from the hundreds column $X=4$, so from the units column $V=7$.
Thus the unique solution is $497+447=944$.