Thread: remainders over divisions

Hey guys, if i had a question of say:

QUESTION: find the remainders when the following numbers are divided by 89.

i)

ii)

is there an easy was to work this out?

thanks!

First note that .

It is easy to see that 10 * 89 = 809 + 81. Move 81 to the left side and consider both sides modulo 89.

If you represent 800 as , then . By the Binomial theorem (or just by expanding) one sees that all terms have the factor 89 except for the last one, namely, .