# The Impossibly Long... Blackboard

**Number Theory**Level 5

On this board, I write out the decimal representation of all the integers from 1 to \(10^{100}\), and it takes \(0.01\) seconds to write each digit; the total time in seconds is \(T\).

I then choose two numbers at random \(a\) and \(b\) and replace both with a single number \(a+ab+b\). After some number of iterations, the single number left is of the form \((n+1)!-1\).

Find

\[\lfloor T \rfloor \pmod{\log_{10}(n)}\]