Multiple (mathematics)

In mathematics, 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.

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.

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:
 * $$ 14 = 7 \times 2;$$
 * $$ 49 = 7 \times 7;$$
 * $$ -21 = 7 \times (-3);$$
 * $$  0 = 7 \times 0;$$
 * $$  3 = 7 \times (3/7), \quad 3/7$$ is not an integer;
 * $$ -6 = 7 \times (-6/7), \quad -6/7$$ is not an integer.

Properties

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

Submultiple
In some texts, "a is a submultiple of b" has the meaning of "a being a unit fraction of b" (a=1/b) or, equivalently, "b being an integer multiple n of a" (b=na). This terminology is also used with units of measurement (for example by the BIPM and NIST ), 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. As another example, one inch may be considered as a 12-fold submultiple of a foot, or a 36-fold submultiple of a yard.