725. The number $11 \ldots 122 \ldots 2$ (written with 100 ones and 100 twos) can be expressed as the product of two consecutive natural numbers.
$\triangle$ First, let's try this for the numbers 12 and 1122:

$$
12=3 \cdot 4, \quad 1122=33 \cdot 34
$$

Apparently, in our case, one of the factors is written entirely with threes, and the other is one more than that.

Let's factorize the given number. But how do we get a factor written entirely with threes? We will have:

$$
\begin{gathered}
11 \ldots 122 \ldots 2=11 \ldots 1 \cdot 10^{100}+2 \cdot 11 \ldots 1=11 \ldots 1 \cdot\left(10^{100}+2\right)= \\
=(11 \ldots 1 \cdot 3) \frac{100 \ldots .02}{3}=33 \ldots 3 \cdot 33 \ldots 34 . \mathbf{\Delta}
\end{gathered}
$$