Separate continuous payoff time from monthly rounding

$M^{\prime}=kM-500$. At $M=10000$, $0=10000k-500$, so $k=1/20$ per month and $M^{\prime}=(M-10000)/20$.

$\int dM/(M-10000)=\int dt/20$ gives $\ln|M-10000|=t/20+C$. Hence $M=10000+Ae^{t/20}$, where $A$ may be negative. The initial value $M(0)=8000$ gives $A=-2000$, so $M=10000-2000e^{t/20}$. The physical loan model stops when this expression first reaches zero; it does not continue into negative debt.

$M=0$ gives $e^{t/20}=5$, hence $t=20\ln5\approx32.188758$ months. Starting on 1 March 2013, 32 calendar months leads to 1 November 2015, so the continuous model clears during November under the usual calendar-month interpretation. The source reports 1 December 2015, the first monthly boundary after rounding up to 33 months. That printed date is retained as the source convention, not as the exact continuous payoff date. The problem does not specify how to convert a fractional month into an exact calendar day.