Dayna writes the integers $$1,2,3,4,5,6,7,8,9,10,11,12$$on a chalkboard, then she erases the integers from $1$ through $6$, as well as their multiplicative inverses $\pmod{13}$. What is the only integer Dayna does not erase?
I notice that the multiplicative inverses of $1,2,3,4,5,6$ are $1,7,9,10,8,11$ respectively, since those are the numbers that multiply with them to give $1 \pmod{13}$. So, Dayna erases $1,2,3,4,5,6,7,8,9,10,11$, which leaves only $12$ on the board. Therefore, the only integer Dayna does not erase is $12$. 