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 ${\displaystyle b/a}$ 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.

In some texts, "a is a submultiple of b" has the meaning of "a being a unit fraction of b" or, equivalently, "b being an integer multiple of a". This terminology is also used with units of measurement (for example by the BIPM[2] and NIST[3]), 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.[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.