TITLE: What will be the 2014th digit?
QUESTION [5 upvotes]: Let $a$ be an integer $\underbrace{33\ldots 33}_{2013}$ and let $b$ be the integer $\underbrace{66\ldots66}_{2013}$
What is the 2014th digit (from the right) that appears in the product $ab$?

I tried using mod, but is it the correct way to approach this problem?

REPLY [9 votes]: \begin{align}
ab&=18\cdot[\frac{10^{2013}-1}{9}]^2=\frac29[10^{4026}-2\cdot10^{2013}+1]\\
&=\frac29[(10^{4026}-1)-2(10^{2013}-1)]\\
&=2\cdot\frac{10^{4026}-1}9-4\cdot\frac{10^{2013}-1}9\\
&=\underbrace{22\ldots 2}_{4026}-\underbrace{44\ldots 4}_{2013}\\
&=\underbrace{22\ldots 2}_{2013}\underbrace{00\ldots 0}_{2013}-\underbrace{22\ldots 2}_{2013}\\
&=\underbrace{22\ldots 2}_{2012}1\underbrace{77\ldots 7}_{2012}8
\\
\end{align}