Prove that the number is the product of two successive positive integers Prove that the number $\underbrace{11\cdots 1}_{100\text{ digits}}\underbrace{22\cdots 2}_{100\text{ digits}}$ is the product of two successive positive integers.
What is the general method for this class of proof (big integers)?

Hint:
$$
\begin{align}
\underbrace{11\cdots 1}_{100\text{ digits}}\underbrace{22\cdots 2}_{100\text{ digits}} & = 1 \cdot 10^{199} + 1 \cdot 10^{198} + \cdots + 1 \cdot 10^{100} + 2 \cdot 10^{99} + 2 \cdot 10^{98} + \cdots + 2 \cdot 10^0 \\
 & = 2 \sum_{k=0}^{99} 10^k + \sum_{k=100}^{199} 10^k \\
 & = \frac{(10^{100}-1)(10^{100}+2)}{9}
\end{align}
$$

[ EDIT ]  To conclude the proof, just note that $\frac{10^{100}-1}{3}$ is an integer, and $\frac{10^{100}+2}{3} = \frac{10^{100}-1}{3} + 1$.
