Multiple (mathematics)

In mathematics, a multiple is the product of any quantity and an integer.^[1] In other words, for the quantities a and b, it can be said that b is a multiple of a if b = na for some integer n, which is called the multiplier. If a is not zero, this is equivalent to saying that ${\displaystyle b/a}${\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.

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 14=7\times 2;}

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

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

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

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

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

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

In some texts^{[which? ]}, "a is a submultiple of b" has the meaning of "a being a unit fraction of b" (a=b/n) 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)

• Arithmetic
• Multiplication
• Integers

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