24. A positive integer is called frierdly if it is divisible by the sum of its digits. For example, 111 is friendly but 123 is not. Find the number of all two-digit friendly numbers.
24. Answer: 23.
Since $\frac{10 a+b}{a+b}=\frac{9 a}{a+b}+1$, we have $(a+b) 9 a$.
Case 1: If $3 \nmid(a+b)$, then $(a+b) \quad a$, and thus $b=0$. We have $1020 \quad 40507080$.
Case 2: If $3(a+b)$ but $9 \nmid(a+b)$, then $(a+b) \quad 3 a$ and $1 \leq \frac{3 a}{a+b} \leq 3$. If $\frac{3 a}{a+b}=1$, then $2 a=b$, we have 1224 48. If $\frac{3 a}{a+b}=2$, then $a=2 b$, we have 214284 . If $\frac{3 a}{a+b}=3$, then $b=0$, we have 3060 .
Case 3: If $9(a+b)$, then $a+b=9$ or 18 . If $a+b=18$, then $a=b=9$, which is impossible. If $a+b=9$, then we have 9 friendly numbers $18 \quad 2736 \quad 455463 \quad 728190$.
Therefore, in total there are 23 2-digit friendly numbers.