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Version 24 |
| 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 |
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} |
\begin{itemize} |
| \item $a\mid f$ and $b\mid f$, |
\item $a\mid f$ and $b\mid f$, |
| \item if $a\mid c$ and $b\mid c$, then $f\mid c$. |
\item if $a\mid c$ and $b\mid c$, then $f\mid c$. |
| \end{itemize} |
\end{itemize} |
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| \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.) |
\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.) |
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| \subsection*{Properties} |
\subsection*{Properties} |
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| \item If \,$a = \prod_{i=1}^{m}p_i^{\alpha_i}$\, and |
\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 |
\,$b = \prod_{i=1}^{m}p_i^{\beta_i}$\, are the prime factor \PMlinkescapetext{presentations} of the positive integers $a$ and $b$ ($\alpha_{i} \geqq 0$, \,$\beta_{i} \geqq 0$ \,$\forall i$), then |
| $$\lcm(a,\,b)= \prod_{i=1}^{m}p_i^{\max\{\alpha_i,\,\beta_i\}}.$$ |
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$$\lcm(a,\,b)= \prod_{i=1}^{m}p_i^{\max(\alpha_i, \beta_i)}.$$ |
| This can be generalized for $\lcm$ of several numbers. |
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 |
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\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 |
| $$\gcd(a,\,b)\cdot \lcm(a,\,b) = ab.$$ |
$$\gcd(a,\,b)\cdot \lcm(a,\,b) = ab.$$ |
| This formula is sensible only for two integers; it can not be generalized for several numbers, i.e., for example, |
This formula is sensible only for two integers; it can not be generalized for several numbers, i.e., for example, |
| $$\gcd(a,\,b,\,c)\cdot \lcm(a,\,b,\,c) \neq abc.$$ |
$$\gcd(a,\,b,\,c)\cdot \lcm(a,\,b,\,c) \neq abc.$$ |
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| \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 |
\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 |
| $$((a)+(b))((a)\cap(b)) = (a)(b).$$ |
$$((a)+(b))((a)\cap(b)) = (a)(b).$$ |
| \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 the {\em Jensen's formula} |
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\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. |
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$R$ is a Pr\"ufer ring iff the {\em Jensen's formula} |
| $$(\mathfrak{a}+\mathfrak{b})(\mathfrak{a}\cap\mathfrak{b}) = \mathfrak{ab}$$ |
$$(\mathfrak{a}+\mathfrak{b})(\mathfrak{a}\cap\mathfrak{b}) = \mathfrak{ab}$$ |
| is true for all ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, with at least one of them having non-zero-divisors. |
is true for all ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, with at least one of them having non-zero-divisors. |
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| \end{enumerate} |
\end{enumerate} |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{Larsen & McCarthy} M. Larsen and P. McCarthy: {\em Multiplicative theory of ideals}. Academic Press. New York (1971). |
\bibitem{Larsen & McCarthy} M. Larsen and P. McCarthy: {\em Multiplicative theory of ideals}. Academic Press. New York (1971). |
| \end{thebibliography} |
\end{thebibliography} |
