Remainders of Expressions

A key principle of modular arithmetic is that "the algebra of the remainders is equal to the remainder of the algebra." Instead of calculating the result of a large expression and then finding the remainder, you can find the remainder of each term first and then perform the algebra on those smaller numbers.

For $(43 - 15 \times 43) \div 6$:
1. Find remainders mod 6: $43 \equiv 1 \pmod{6}$ and $15 \equiv 3 \pmod{6}$.
2. Perform algebra on remainders: $1 - (3 \times 1) = -2$.
3. Adjust if negative: $-2 \equiv -2 + 6 \equiv 4 \pmod{6}$. The remainder is 4.