View source for Least common multiple

The smallest positive number among the common [[multiple]]s of a finite set of integers or, in particular, of natural numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l0577501.png" />. The least common multiple of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l0577502.png" /> exists if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l0577503.png" />. It is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l0577504.png" />.

Properties of the least common multiple are:

1) the least common multiple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l0577505.png" /> is a divisor of any other common multiple;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l0577506.png" />;

3) if the integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l0577507.png" /> are expressed as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l0577508.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l0577509.png" /> are distinct primes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l05775010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l05775011.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l05775012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l05775013.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l05775014.png" /></td> </tr></table>

4) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l05775015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l05775016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l05775017.png" /> is the [[Greatest common divisor|greatest common divisor]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l05775018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l05775019.png" />.

Thanks to the last property, the least common multiple of two numbers can be found with the aid of the [[Euclidean algorithm|Euclidean algorithm]]. The concept of the least common multiple can be defined for elements of an integral domain, and also for ideals of a commutative ring.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,   "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Bukhshtab,   "Number theory" , Moscow  (1966)  (In Russian)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  R. Faure,   A. Kaufman,   M. Denis-Papin,   "Mathématique nouvelles" , '''1–2''' , Dunod  (1964)</TD></TR>
</table>



====Comments====
Other frequently used notations for the least common multiple are: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l05775020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l05775021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057750/l05775022.png" />, etc. In a unique factorization domain least common multiples exist and are unique (up to units).