I was reading a textbook and came across the following approach to find the LCM and GCD of rational numbers/fractions:
Can someone please help me understand why the above formula holds true or how the above is logically deduced?
Notation: $\text{HCF}$ is denoted below as $\text{gcd}$.
Assume you have two fractions $\frac{a}{b},\frac{c}{d}$ reduced to lowest terms. Let
$$\begin{eqnarray*} a &=&\underset{i}{\prod } p_{i}^{e_{i}(a)},\qquad b=\underset{i}{\prod } p_{i}^{e_{i}(b)}, \\ c &=&\underset{i}{\prod } p_{i}^{e_{i}(c)},\qquad d=\underset{i}{\prod } p_{i}^{e_{i}(d)}. \end{eqnarray*}$$
be the prime factorizations of the integers $a,b,c$ and $d$. Then
$$\frac{\underset{i}{\prod }\ p_{i}^{\max \left( e_{i}(a),e_{i}(c)\right) }}{% \prod_{i}\ p_{i}^{\min \left( e_{i}(b),e_{i}(d)\right) }}$$
is a fraction which is a common multiple of $\frac{a}{b},\frac{c}{d}$. It is the least one because by the properties of the $\text{lcm}$ and $\gcd $ of two integers, $\prod_{i}\ p_{i}^{\max \left( e_{i}(a),e_{i}(c)\right) }$ is the least common multiple of the numerators and $\prod_{i}\ p_{i}^{\min \left( e_{i}(b),e_{i}(d)\right) }$ is the greatest common divisor of the denominators. Hence
$$\begin{eqnarray*} \text{lcm}\left( \frac{a}{b},\frac{c}{d}\right) &=&\text{lcm}\left( \frac{{\prod_{i}\ p_{i}^{e_{i}(a)}}}{\prod_{i}\ p_{i}^{e_{i}(b)}},\frac{% \prod_{i}\ p_{i}^{e_{i}(c)}}{\prod_{i}\ p_{i}^{e_{i}(d)}}\right)=\frac{\prod_{i}\ p_{i}^{\max \left( e_{i}(a),e_{i}(c)\right) }}{% \prod_{i}\ p_{i}^{\min \left( e_{i}(b),e_{i}(d)\right) }}=\frac{\text{lcm}(a,c)}{\gcd (b,d)}.\quad(1) \end{eqnarray*}$$
Similarly
$$\begin{eqnarray*} \gcd \left( \frac{a}{b},\frac{c}{d}\right) =\gcd \left( \frac{{\prod_{i}\ p_{i}^{e_{i}(a)}}}{\prod_{i}p_{i}^{e_{i}(b)}},\frac{% \prod_{i\ }p_{i}^{e_{i}(c)}}{\prod_{i}p_{i}^{e_{i}(d)}}\right) =\frac{\prod_{i}\ p_{i}^{\min \left( e_{i}(a),e_{i}(c)\right) }}{% \prod_{i}\ p_{i}^{\max \left( e_{i}(b),e_{i}(d)\right) }} =\frac{\gcd (a,c)}{\text{lcm}(b,d)}.\quad(2) \end{eqnarray*}$$
The repeated application of these relations generalizes the result to any finite number of fractions.
This is how I understand it,
Let $\dfrac 23$, $\dfrac 56$ and $\dfrac 79$ be three fractions. Let $\dfrac AB$ be their LCM. Now all the fractions must divide $\dfrac AB$, that is,$ \dfrac{A}{B} ÷ \dfrac{2}{3},\dfrac{A}{B} ÷ \dfrac{5}{6}$ and $\dfrac{A}{B} ÷ \dfrac{7}{9} $ all must be natural numbers. Or $\dfrac{A\cdot3}{B\cdot2}, \dfrac{A\cdot6}{B\cdot5}$, and $\dfrac{A\cdot9}{B\cdot7}$ all must be natural numbers. This requires that $A$ must be divisible by $2,5$ and $7$ each and $B$ must be a factor of $3,6$ and $9$ each. In other words, $A$ must be a multiple of $2,5$, and $7$. Now $\dfrac{A}{B}$ must be the lowest possible value, which requires us to choose the highest possible value of $B$ (HCF of denominators) and the lowest possible value of A (LCM of numerators).
Similarly, we can understand how to find the HCF of a given number of fractions.
Below is a proof using $\,\rm\color{#0a0}{E} =$ Euclid's Lemma and $\,\rm\color{#c00}{U} = $ universal definitions of GCD, LCM.
Theorem $\ \ \gcd\left(\dfrac{a_1}{b_1},\ldots,\dfrac{a_n}{b_n}\right) = \dfrac{\gcd(a_1,\ldots,a_n)}{{\rm lcm}(b_1,\ldots,b_n)}\,\ $ if $\,\ \gcd(a_i,b_i)\!=\!1\,$ for all $\,i$
$\begin{align}{\bf Proof}\ \ \ {\rm Reduced}\ \ \smash{\dfrac{c}d\,{\Large \mid}\,\dfrac{a_i}{b_i}},\,\ \forall\, i \iff\ & c b_i\mid da_i,\,\ \forall\, i\\ \overset{\rm\color{#0a0}E\!}\iff\ & c\mid a_i,\,\ b_i\mid d,\,\ \forall\, i,\ \ {\rm by}\,\ (c,d)\!=\!1\!=\!(a_i,b_i)\\ \overset{\rm\color{#c00}U}\iff\ & c\mid \gcd\{a_i\},\,\ {\rm lcm}\{b_i\}\mid d\\ \overset{\rm\color{#0a0}E\!}\iff\ & c\ {\rm lcm}\{b_i\}\:\!\!\mid d\:\!\gcd\{a_i\},\, \ {\rm by}\,\ (c,d)\!=\!1\!=\!(a_i,b_i)\\[.4em] \iff\ &\! \frac{c}d\:{\Large \mid}\:\frac{\gcd\{a_i\}}{{\rm lcm}{\{b_i\}}} \end{align}$
Remark $ $ The proof for LCM is the dual of the above proof (which works in any GCD domain). These ideas go back to Euclid, who defined the greatest common measure of line segments. Nowadays this can also be viewed in terms of fractional ideals or Krull's $v$-ideals.