How does modular arithmetic get used in abstract algebra?

My answer to a question on Quora: How does modular arithmetic get used in abstract algebra?

Perhaps this is the simplest answer: the rules of signs in multiplication:

$$“\boldsymbol{+}” \;\times\; “\boldsymbol{+}” \quad=\quad “\boldsymbol{+}”;\qquad “\boldsymbol{-}”\; \times\; “\boldsymbol{-}”\quad =\quad “\boldsymbol{+}”;$$

$$“\boldsymbol{+}”\; \times\; “\boldsymbol{-}”\quad = \quad“\boldsymbol{-}”; \qquad“\boldsymbol{-}”\; \times \;“\boldsymbol{+}” \quad = \quad“\boldsymbol{-}”;$$

(that is , “plus times minus is plus”, “minus times minus is is plus”, etc.)

are the same as addition in modular arithmetic modulo 2: replace $$“\boldsymbol{+}”$$ by 0, $$“\boldsymbol{-}”$$ by 1, and $$“\times”$$ by $$“+”$$. This basic observation is already surprisingly useful.

Consider this elementary problem:

We are given 101 coins such that after removal of any one of them the remaining 100 coins can be redistributed in two groups of 50 coins in such a way that the sum of weights of coins in each group is the same. Prove that all coins have equal weights.

I know three different solutions (some use serious, by undergraduate standards, abstract algebra), and I’ll publish them soon on my blog/journal “Selected Passages From Correspondence With Friends , under the title “101 Coins” — check it in a couple of days. All three use, in various ways, and sometimes in disguise, modular arithmetic modulo 2.