Multiple (mathematics)

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.

• 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 ${\displaystyle x}$.

In some texts, "a is a submultiple of b" has the meaning of "a being a unit fraction of b" (a=1/b) or, equivalently, "b being an integer multiple n of a" (b=n a). This terminology is also used with units of measurement (for example by the BIPM[2] and NIST[3]), where a unit submultiple is obtained by prefixing the main unit, defined as the quotient of the main unit by an integer, mostly a power of 10³. For example, a millimetre is the 1000-fold submultiple of a metre.[2][3] As another example, one inch may be considered as a 12-fold submultiple of a foot, or a 36-fold submultiple of a yard.

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