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$N$ is a 3 digit positive integer. It can be represented in base 5 and base 6 as the strings $n_5$ and $n_6$. If we then treat $n_5$ and $n_6$ as the base 10 encodings of two integers $N_5$ and $N_6$, we find that the sum $N_5 + N_6$ has the same two rightmost digits as $2N$.

How many possible choices do we have for $N$?

This question is from the AMC 10B 2013, problem 25. You can find the original problem statement here:

My attempt at this problem led me to an equation evolving the modulo operator that I could not solve by hand. I ask for strategies to solve such equations here: Solving equations involving modulo operator

I eventually gave up and read the solution posted on the website, but it made sense only up to the halfway mark, when the writer employs an argument about treating a number as a units digit instead of a tens digit.

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