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Multiple (mathematics)

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 Title: Multiple (mathematics) Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

Multiple (mathematics)

In mathematics, 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 or coefficient. If a is not zero, this is equivalent to saying that b/a is an integer with no remainder.[4][5][6] 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.

References

1. ^ Weisstein, Eric W., "Multiple", MathWorld.
2. ^ WordNet lexicon database, Princeton University
3. ^ WordReference.com
4. ^ The Free Dictionary by Farlex
5. ^ Dictionary.com Unabridged
6. ^ Cambridge Dictionary Online