Multiple (mathematics)

In mathematics, a multiple is the product of any quantity and an integer. 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 or coefficient. If a is not zero, this is equivalent to saying that b/a is an integer with no remainder. If a and b are both integers, and b is a multiple of a, then a is called a divisor of b.

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:

• $14 = 7 \times 2$
• $49 = 7 \times 7$
• $-21 = 7 \times (-3)$
• $0 = 7 \times 0$
• $3 = 7 \times (3/7)$, $3/7$ is a rational number, not an integer
• $-6 = 7 \times (-6/7)$, $-6/7$ is a rational number, not an integer.

Properties

• 0 is a multiple of everything ($0=0\cdot b$).
• 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 $x$.

See also

• Ideal (ring theory)
• Decimal and SI prefix