<s>[INST] Prove that number $\underbrace{11 \ldots1}_{100} \underbrace{22 \ldots2}_{100}$ is product of two consecutive numbers Prove that number $\underbrace{11 \ldots1}_{100}$$\underbrace{22 \ldots2}_{100}$ is product of  two consecutive numbers
$\begin{align}\underbrace{11 \ldots1}_{100} \underbrace{22 \ldots2}_{100}&=10^{199}+10^{198}+\ldots+10^{100}+2(10^{99}+10^{98}+\ldots+10+1)\\&=(10^{100}+2)(10^{99}+10^{98}+\ldots+10+1)=(10^{100}+2)\frac{10^{100}-1}{10-1}\end{align}$.  
Is this good path or not?
 [/INST] Insert $1=\frac{1}{3}*3$
$$
(10^{100}+2)\frac{1}{3} 3\frac{10^{100}-1}{10-1}\\
x= \frac{10^{100}+2}{3}\\
y=3\frac{10^{100}-1}{10-1} = \frac{10^{100}-1}{3}\\\\
$$
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