jmc

Notice that $100\equiv -1(\mathrm{mod}101)$. Therefore $$120000\equiv -1200\equiv 12(\mathrm{mod}101).$$Likewise $$3400\equiv -34(\mathrm{mod}101).$$Combining these lets us write $$123456\equiv 12-34+56(\mathrm{mod}101)$$or $$123456\equiv \boxed{34}(\mathrm{mod}101).$$