Multiple (mathematics)

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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 old texts, "a is a submultiple of b" was equivalent with "b is a multiple of a".[7][8] This terminology is no more in use, except for units of measurement, where a submultiple of a main unit is a unit, named by prefixing the main unit, which is a quotient of the main unit by an integer, generally a power of 10. For example, a millimetre is a submultiple of a metre.[9]


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:

  • , is a rational number, not an integer
  • , is a rational number, not an integer.


  • 0 is a multiple of everything ().
  • The product of any integer and any integer is a multiple of . In particular, , which is equal to , is a multiple of (every integer is a multiple of itself), since 1 is an integer.
  • If and are multiples of then and are also multiples of .


  1. ^ Weisstein, Eric W. "Multiple". MathWorld. 
  2. ^ WordNet lexicon database, Princeton University
  3. ^
  4. ^ The Free Dictionary by Farlex
  5. ^ Unabridged
  6. ^ Cambridge Dictionary Online
  7. ^ "Submultiple". Merriam-Webster Online Dictionary. Merriam-Webster. 2017. Retrieved February 1, 2017. 
  8. ^ "Submultiple". Oxford Living Dictionaries. Oxford University Press. 2017. Retrieved February 1, 2017. 
  9. ^ "NIST Guide to the SI".  Section 4.3: Decimal multiples and submultiples of SI units: SI prefixes

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