# Does It Perpetually Hold?

**Number Theory**Level 2

\[ \begin{array} {r c r c l }
0.1 &\times & 0.1 & = &0.01 \\
0.11 & \times& 0.11 & = &0.012\ldots \\

0.111 & \times& 0.111 & =& 0.0123\ldots \\

& & & \vdots& \\

0.\underbrace{1111\ldots1}_{\text{nine }9\text{'s}} & \times&

0.\underbrace{1111\ldots1}_{\text{nine }9\text{'s}} & =& 0.0123456789\ldots \\
0.\underbrace{1111\ldots1}_{\text{ten }9\text{'s}} & \times&

0.\underbrace{1111\ldots1}_{\text{ten }9\text{'s}} & =& 0.012345678910\ldots \\
\end{array} \]

Exactly how many of the equations above are true?

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