Multiple (mathematics)

Short description: Product with an integer

In science, a multiple is the product of any quantity and an integer. 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 b/a is an integer.

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". This terminology is also used with units of measurement (for example by the BIPM and NIST), 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. 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:

• $\displaystyle{ 14 = 7 \times 2 }$
• $\displaystyle{ 49 = 7 \times 7 }$
• $\displaystyle{ -21 = 7 \times (-3) }$
• $\displaystyle{ 0 = 7 \times 0 }$
• $\displaystyle{ 3 = 7 \times (3/7), \quad 3/7 }$ is not an integer
• $\displaystyle{ -6 = 7 \times (-6/7), \quad -6/7 }$ is not an integer.

Properties

• 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 $\displaystyle{ x }$.