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Look up multiple or submultiple in Wiktionary, the free dictionary.

In science, a multiple is the product of any quantity and an integer.^[1][2][3] In other words, for the quantities a and b, we say 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 b/a is an integer.^[4][5][6]

In mathematics, 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.

In some texts, "a is a submultiple of b" has the meaning of "b being an integer multiple of a".^[7][8] This terminology is also used with units of measurement (for example by the BIPM^[9] and NIST^[10]), where a submultiple of a main unit is a unit, named 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.^[9][10] As another example, one inch may be considered as a 12-fold submultiple of a foot, or a 36-fold submultiple of a yard.

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:

• 14=7×2displaystyle 14=7times 2


• 49=7×7displaystyle 49=7times 7


• −21=7×(−3)displaystyle -21=7times (-3)


• 0=7×0displaystyle 0=7times 0


• 
  3=7×(3/7),3/7displaystyle 3=7times (3/7),quad 3/7 is a rational number, not an integer


• 
  −6=7×(−6/7),−6/7displaystyle -6=7times (-6/7),quad -6/7 is a rational number, not an integer.

Properties

• 0 is a multiple of everything (0=0⋅bdisplaystyle 0=0cdot b).


• The product of any integer ndisplaystyle n and any integer is a multiple of ndisplaystyle n. In particular, ndisplaystyle n, which is equal to n×1displaystyle ntimes 1, is a multiple of ndisplaystyle n (every integer is a multiple of itself), since 1 is an integer.


• If adisplaystyle a and bdisplaystyle b are multiples of xdisplaystyle x then a+bdisplaystyle a+b and a−bdisplaystyle a-b are also multiples of xdisplaystyle x.

References

See also

• Unit fraction


• Ideal (ring theory)


• 
  Decimal and SI prefix
  


• Multiplier (linguistics)