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''of a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065340/m0653401.png" />''
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''of a natural number $n$''
  
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A [[Natural number|natural number]] that is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065340/m0653402.png" /> without remainder (cf. [[Division|Division]]). A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065340/m0653403.png" /> divisible by each of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065340/m0653404.png" /> 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 common multiple) and the others are multiples of the lowest common multiple. If the [[Greatest common divisor|greatest common divisor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065340/m0653405.png" /> of two numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065340/m0653406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065340/m0653407.png" /> is known, the lowest common multiple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065340/m0653408.png" /> is found from the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065340/m0653409.png" />.
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A [[natural number]] that is divisible by $n$ without remainder (cf. [[Division]]). A number $n$ divisible by each of the numbers $a,b,\ldots,m$ 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 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 $m = ab/d$.
  
 
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.M. Vinogradov,  "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian)</TD></TR></table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  I.M. Vinogradov,  "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian 5th ed. 1949) {{ZBL|0057.28201}}</TD></TR>
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Revision as of 18:20, 16 January 2016

of a natural number $n$

A natural number that is divisible by $n$ without remainder (cf. Division). A number $n$ divisible by each of the numbers $a,b,\ldots,m$ 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 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

References

[a1] I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian 5th ed. 1949) Zbl 0057.28201
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Multiple. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiple&oldid=37547
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