# 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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