Divisibility Rules: The Complete Guide
Big Idea: Divisibility rules are shortcuts that let you determine whether one integer divides evenly into another without performing long division. They connect number theory to algebra and are essential tools for simplifying fractions, factoring expressions, and working with prime factorization.
Formal Definition
An integer $a$ is divisible by a nonzero integer $b$ if there exists an integer $k$ such that:
$$a = b \times k$$
We write $b \mid a$ (read: "$b$ divides $a$"). Equivalently, $a \div b$ has a remainder of $0$, or $a \mod b = 0$.