# Multiple

From Encyclopedia of Mathematics

*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://encyclopediaofmath.org/index.php?title=Multiple&oldid=39606