How can multiplication be performed with congruence classes modulo 5?

In modular arithmetic, multiplication of congruence classes modulo 5 is performed by selecting any representative integer from each class, multiplying those integers together, and then determining the resulting congruence class modulo 5.

To multiply two classes, follow these steps:
1. Choose any element from the first congruence class and any element from the second.
2. Multiply these two integers using standard arithmetic.
3. Find the remainder of that product when divided by 5. This remainder identifies the resulting congruence class.

For example, to multiply the class containing $-3$ by the class containing $4$ modulo 5:
* Multiply the representatives: $-3\times 4=-12$.
* Determine the congruence: Since $-12$ divided by 5 leaves a remainder of 3 (because $-12=5\times (-3)+3$), the product is congruent to 3 modulo 5.

This method is consistent regardless of which representatives are chosen from the respective classes, as the operation is well-defined within the system of residue classes.