10. The sum of the digits of the positive integer $N$ is three times the sum of the digits of $N+1$. What is the smallest possible sum of the digits of $N$ ?
A 9
B 12
C 15
D 18
E 27
Solution
B

The sum of the digits of a number decreases only if the number has the last digit 9 , so that the next number ends in 0 , and 1 is added to its penultimate digit. In that case, the sum of the digits of the number $N+1$ decreases by 8 , compared to that of the number $N$. At the same time, it is one third of the sum of the digits of $N$. Denoting the sum of the digits of $N$ by $x$, we have $x=3(x-8)$, which gives $x=12$.

Note that if the number ends in more that one 9 , its sum of digits will be at least 19 , which is already greater than 12 .
The number 39 has the property described above and has a sum of digits of 12 .