Formatting of numbers while finding remainder using Binomial theorem

Question

While Dividing $$2^{501}$$ by 21 to find remainder why do we need to write it in the form of $$8(63+1)^{83}$$ and not like $$2(21-5)^{125}$$ like why should it always be in the form of (1+x) ?

Answer 1

If you do it the second way, it is easy to see that $$2(21-5)^{125}\equiv-2\times5^{125}\pmod{21}\ .$$ But this still leaves you a fair bit of work to do. If you do it the first way, you get the final answer almost immediately.