Remainders with a/p, b/p, and ab/p

These problems test the principle that "the algebra of remainders equals the remainder of the algebra." You are given the remainders of expressions involving variables and asked to find the remainder of a related expression.

If $\frac{6x}{7}$ has a remainder of 2 and $\frac{2y}{7}$ has a remainder of 3, to find the remainder of $\frac{4xy}{7}$: Multiply the two initial expressions: $\frac{12xy}{7}$. The product of their remainders is $2 \times 3 = 6$. So, $\frac{12xy}{7}$ has a remainder of 6. Since $\frac{4xy}{7}$ is one-third of this, the remainder is $6 \div 3 = 2$.