15. Find the smallest three-digit number $n$ such that if the three digits are $a, b$ and $c$, then
$$
n=a+b+c+a b+b c+a c+a b c .
$$
15. Ans: 199
From $100 a+10 b+c=a+b+c+a b+b c+a c+a b c$, we get $(99-b-c-b c) a=b(c-9)$. Note that LHS $\geq 0$ (since $b+c+b c=(1+b)(1+c)-1 \leq 99)$ and RHS $\leq 0$. Thus we must have $b=c=9$. Since $a \geq 1, n=199$ is the answer.