Multiple
A multiple of a number is the product of that number and any positive integer.
Definition
A multiple of a number is what you get when you multiply it by $1$, $2$, $3$, $4$, and so on, the numbers in a times table; formally the multiples of $n$ are the set $\{n, 2n, 3n, 4n, \ldots\} = \{kn : k \text{ is a positive integer}\}$, an arithmetic sequence with common difference $n$, and $n$ is a multiple of $d$ if and only if $d$ is a factor of $n$. In $\mathbb{Z}$, the set $n\mathbb{Z}$ is the principal ideal generated by $n$, the subgroup of $(\mathbb{Z},+)$ generated by $n$, and two integers are congruent modulo $n$ if their difference is a multiple of $n$.
Example
Multiples of $4$ are $4, 8, 12, 16, 20, 24, \ldots$, the results of $4\times1, 4\times2, 4\times3, \ldots$; every multiple of $4$ can be divided exactly by $4$. Common multiples of $4$ and $6$ are $12, 24, 36, \ldots$, the smallest being $\text{LCM}(4,6) = 12$, and all common multiples are multiples of the LCM. The intersection of $4\mathbb{Z}$ and $6\mathbb{Z}$ is $12\mathbb{Z}$ (multiples of $\text{LCM}(4,6)=12$), while the sum $4\mathbb{Z} + 6\mathbb{Z} = 2\mathbb{Z}$ (multiples of $\text{GCD}(4,6)=2$), illustrating Bezout's identity.
Key Insight
Every multiple of $4$ can be divided exactly by $4$, and multiples and factors are opposites: if $4$ is a factor of $20$, then $20$ is a multiple of $4$. The set of all multiples of $n$ (including negative integers and zero) forms an ideal in the ring $\mathbb{Z}$, and LCM and GCD correspond to the intersection and sum of ideals respectively, an algebraic interpretation that unifies arithmetic with ring theory and explains why $\text{LCM} \times \text{GCD} = a \times b$ for positive integers $a$ and $b$.