Let the three digit number be a*100 + b*10 +c*1, where a,b and c are digits and a is not 0.
The sum of the digits are a+b+c and their product is abc
We want a+b+c = abc
Since a is not 0, a+b+c is not 0 (it is at least 1), so abc is not 0 which means none of a, b and c are 0.
Now the smallest number we can use is 111. Now 11x will not work as the sum is x+2 and the product is x
What about 12X? The sum is 3+x and the product is 2x. Now if 3+x=2x, then x=3. So the smallest number with the desired property is 123
