Question:
The rightmost non-zero digit in  \begin{align*}
&(1001001)(1010101)+(989899)(1001001)\\
&\qquad -(1001)(989899)-(1010101)(1001)
\end{align*}is $a$, and it is followed by $b$ zeroes. Find the ordered pair $(a,b)$.

Answer:
We can factor the given product using Simon's Favorite Factoring Trick. Factor $1001001$ out of the first two terms and $-1001$ out of the second two terms to find $$(1001001)(1010101+989899)-1001(989899+1010101).$$Since $1010101+989899=2000000$, we can complete the factoring as \begin{align*}(1001001-1001)(2000000)&=(1000000)(2000000)\\&=2000000000000.\end{align*}Thus we can see that the rightmost non-zero digit $a=2$, and it is followed by 12 zeroes so $b=12$. Thus $(a,b)=\boxed{(2,12)}$.