least common multipleLeastCommonMultiple2004-03-19 15:46:012007-06-01 02:25:19Definitiongreatest common divisor% this is the default PlanetMath preamble. as your knowledge
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\DeclareMathOperator{\lcm}{lcm}If $a$ and $b$ are two positive integers, then their {\em least common multiple}, denoted by $\lcm(a,\,b)$, is the positive integer $f$ satisfying the conditions
\begin{itemize}
\item $a\mid f$ and $b\mid f$,
\item if $a\mid c$ and $b\mid c$, then $f\mid c$.
\end{itemize}
\textbf{Note:} \, The definition can be generalized for several numbers. \,The positive $\lcm$ of positive integers is uniquely determined. (Its negative satisfies the same two conditions.)
\subsection*{Properties}
\begin{enumerate}
\item If \,$a = \prod_{i=1}^{m}p_i^{\alpha_i}$\, and
\,$b = \prod_{i=1}^{m}p_i^{\beta_i}$\, are the prime factor \PMlinkescapetext{presentations} of the positive integers $a$ and $b$ ($\alpha_{i} \geqq 0$, \,$\beta_{i} \geqq 0$ \,$\forall i$), then
$$\lcm(a,\,b)= \prod_{i=1}^{m}p_i^{\max\{\alpha_i,\,\beta_i\}}.$$
This can be generalized for $\lcm$ of several numbers.
\item Because the greatest common divisor has the expression \,$\gcd(a,\,b) = \prod_{i=1}^{m}p_i^{\min\{\alpha_i,\,\beta_i\}}$, we see that
$$\gcd(a,\,b)\cdot \lcm(a,\,b) = ab.$$
This formula is sensible only for two integers; it can not be generalized for several numbers, i.e., for example,
$$\gcd(a,\,b,\,c)\cdot \lcm(a,\,b,\,c) \neq abc.$$
\item The preceding formula may be presented in \PMlinkescapetext{terms} of ideals of $\mathbb{Z}$; we may replace the integers with the corresponding principal ideals. \,The formula acquires the form
$$((a)+(b))((a)\cap(b)) = (a)(b).$$
\item The recent formula is valid also for other than principal ideals and even in so general systems as the Pr\"ufer rings; in fact, it could be taken as defining property of these rings: \, Let $R$ be a commutative ring with non-zero unity. \,$R$ is a Pr\"ufer ring iff {\em Jensen's formula}
$$(\mathfrak{a}+\mathfrak{b})(\mathfrak{a}\cap\mathfrak{b}) = \mathfrak{ab}$$
is true for all ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, with at least one of them having \PMlinkname{non-zero-divisors}{ZeroDivisor}.
\end{enumerate}
\begin{thebibliography}{9}
\bibitem{Larsen & McCarthy} M. Larsen and P. McCarthy: {\em Multiplicative theory of ideals}. Academic Press. New York (1971).
\end{thebibliography}