jmc

Notice that $100\equiv -1(\mathrm{mod}101)$. Therefore $$310000\equiv -3100\equiv 31(\mathrm{mod}101).$$Likewise $$4100\equiv -41(\mathrm{mod}101).$$Combining these lets us write $$314159\equiv 31-41+59\equiv 49(\mathrm{mod}101).$$However we started with a negative. We actually want to compute $$-314159\equiv -49\equiv \boxed{52}(\mathrm{mod}101).$$