# Multiple

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of a natural number $n$

A natural number that is the result of multiplication of $n$ by some natural number; hence a number divisible by $n$ without remainder (cf. Division). A number $n$ divisible by each of the numbers $a,b,\ldots,k$ is called a common multiple of these numbers. Among all common multiples of two or more numbers, one (distinct from zero) is the smallest (the lowest or least common multiple) and the others are then multiples of the lowest common multiple. If the greatest common divisor $d$ of two numbers $a$ and $b$ is known, the lowest common multiple $m$ is found from the formula $m = ab/d$.

#### Comments

See also Divisibility in rings.

#### References

 [a1] I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian 5th ed. 1949) Zbl 0057.28201
How to Cite This Entry:
Multiple. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Multiple&oldid=39606