Multiple (mathematics)

In

, this is equivalent to saying that ${\displaystyle b/a}$ is an integer.

When a and b are both integers, and b is a multiple of a, then a is called a divisor of b. One says also that a divides b. If a and b are not integers, mathematicians prefer generally to use integer multiple instead of multiple, for clarification. In fact, multiple is used for other kinds of product; for example, a polynomial p is a multiple of another polynomial q if there exists third polynomial r such that p = qr.

Examples

14, 49, −21 and 0 are multiples of 7, whereas 3 and −6 are not. This is because there are integers that 7 may be multiplied by to reach the values of 14, 49, 0 and −21, while there are no such integers for 3 and −6. Each of the products listed below, and in particular, the products for 3 and −6, is the only way that the relevant number can be written as a product of 7 and another real number:

${\displaystyle 14=7\times 2;}$

${\displaystyle 49=7\times 7;}$

${\displaystyle -21=7\times (-3);}$

${\displaystyle 0=7\times 0;}$

${\displaystyle 3=7\times (3/7),\quad 3/7}$ is not an integer;

${\displaystyle -6=7\times (-6/7),\quad -6/7}$ is not an integer.

Properties

• 0 is a multiple of every number (${\displaystyle 0=0\cdot b}$).
• The product of any integer ${\displaystyle n}$ and any integer is a multiple of ${\displaystyle n}$. In particular, ${\displaystyle n}$, which is equal to ${\displaystyle n\times 1}$, is a multiple of ${\displaystyle n}$ (every integer is a multiple of itself), since 1 is an integer.
• If ${\displaystyle a}$ and ${\displaystyle b}$ are multiples of ${\displaystyle x,}$ then ${\displaystyle a+b}$ and ${\displaystyle a-b}$ are also multiples of ${\displaystyle x}$.

Submultiple

In some texts, "a is a submultiple of b" has the meaning of "a being a

.

See also

• Unit fraction
• Ideal (ring theory)
• SI prefix
• Multiplier (linguistics)

References