Modular Arithmetic Product Congruence

Prove that $(a \cdot b) \mod n = (a' \cdot b') \mod n$ when $a \mod n = a'$ and $b \mod n = b'$.

This problem explores the properties of modular arithmetic, a fundamental concept within number theory that is extended into various branches of mathematics and computer science. At its core, modular arithmetic involves operations where numbers wrap around upon reaching a certain value—the modulus. This particular problem focuses on understanding and proving the congruence of products in modular systems. Recognizing the importance of modular arithmetic means understanding how these operations preserve equivalence classes, or congruence classes, based on the modulus. The equivalence relation defined by congruence modulo a number wraps multiple concepts such as divisibility, residue systems, and arithmetic operations into a cohesive understanding.

To approach this proof, one should leverage the properties of congruences and how these properties impact multiplicative relationships. The primary tactic is to express the numbers a and b in terms of their modular equivalents a' and b', and examine how their product behaves similarly under the same modulus. Understanding this concept involves recognizing patterns and properties such as the distributive, associative, and commutative properties within the realm of modular arithmetic. It highlights the elegance of modular calculations which find applications not only in theoretical mathematics but also in cryptographic algorithms and digital systems where such properties are integral.