Multiple
ArithmeticA 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. Multiples are the numbers in a times table.
Example
Multiples of $4$: $4, 8, 12, 16, 20, 24, \ldots$ They are the results of $4\times1, 4\times2, 4\times3, \ldots$
Key Insight
Every multiple of $4$ can be divided exactly by $4$. Multiples and factors are opposites: if $4$ is a factor of $20$, then $20$ is a multiple of $4$.
Definition
The multiples of $n$ are the set $\{n, 2n, 3n, 4n, \ldots\} = \{kn : k \text{ is a positive integer}\}$. In number theory, multiples form an arithmetic sequence with common difference $n$. $n$ is a multiple of $d$ if and only if $d$ is a factor of $n$.
Example
Common multiples of $4$ and $6$: $12, 24, 36, \ldots$ The smallest is the $\text{LCM}(4,6) = 12$. All common multiples are multiples of the LCM.
Key Insight
The set of all multiples of $n$ (including negative integers and zero) forms the ideal $n\mathbb{Z}$ in the ring $\mathbb{Z}$. Ideals generalize the concept of multiples to abstract algebra.
Definition
In $\mathbb{Z}$, the set $n\mathbb{Z} = \{\ldots, -2n, -n, 0, n, 2n, \ldots\}$ is the principal ideal generated by $n$. This is the subgroup of $(\mathbb{Z},+)$ generated by $n$. Two integers are congruent modulo $n$ if their difference is a multiple of $n$. The ideals of $\mathbb{Z}$ are exactly the sets $n\mathbb{Z}$ for non-negative integers $n$.
Example
The ideal $6\mathbb{Z}$ contains all multiples of $6$. The intersection of $4\mathbb{Z}$ and $6\mathbb{Z}$ is $12\mathbb{Z}$ (multiples of $\text{LCM}(4,6)=12$). The sum $4\mathbb{Z} + 6\mathbb{Z} = 2\mathbb{Z}$ (multiples of $\text{GCD}(4,6)=2$), illustrating Bezout's identity.
Key Insight
LCM and GCD correspond to intersection and sum of ideals respectively. This algebraic interpretation unifies arithmetic with ring theory and explains why $\text{LCM} \times \text{GCD} = a \times b$ for positive integers $a$ and $b$.