jmc

We note that $24\equiv -2(\mathrm{mod}13)$ and $15\equiv 2(\mathrm{mod}13)$. We cleverly use these congruences to set up values that vanish in the arithmetic: $$24^{50}-15^{50}\equiv (-2{)}^{50}-2^{50}\equiv 2^{50}-2^{50}\equiv \boxed{0}(\mathrm{mod}13).$$