Divisibility in Multiplicative Structures

If $a$ divides $b$, then $c \times a$ divides $c \times b$.

In this problem, we explore the concept of divisibility, an essential idea in number theory, which is in itself a cornerstone for many areas in discrete mathematics. The problem involves understanding how divisibility relationships change when you scale both the divisor and the dividend by a common factor. Specifically, you're given two integers, a and b, with the property that a divides b. The goal is to demonstrate that if you multiply both a and b by another integer, c, the divisibility relationship is preserved.

The key to understanding this problem lies in the principle that multiplying both terms of a divisibility relationship by the same non-zero integer does not affect the relationship. This is because multiplication by an integer is a linear operation, preserving the inherent properties of the original equation or inequality. Therefore, if a divides b, then ca divides cb because you can factor out c from both terms, reducing it back to the original statement where a divides b.

While this problem may appear simple at first glance, it serves as an excellent illustration of the fundamental properties of numbers under multiplication, and can be extended to solving more complex congruences in modular arithmetic. By fully grasping this foundational understanding, you leverage a powerful tool that can simplify many more advanced mathematical problems and proofs.