Let $m=999 \ldots 99$ be the number formed by 77 digits all equal to 9 and let $n=777 \ldots 77$ be the number formed by 99 digits all equal to 7. What is the number of digits of $m \cdot n$?
Solution

Since $m+1=10^{77}$, notice that:

$$
\begin{aligned}
m \cdot n & =(m+1) \cdot n-n \\
& =\underbrace{777 \ldots 77}_{99} \underbrace{000 \ldots 00}_{77}-\underbrace{777 \ldots 77}_{99} .
\end{aligned}
$$

Since $\underbrace{777 \ldots 77}_{99} \underbrace{000 \ldots 00}_{77}$ has $99+77$ digits and $\underbrace{777 \ldots 77}_{99}$ is less than $\underbrace{777 \ldots 77}_{175}$, the result of the subtraction above will still have 176 digits.