Relevancy to GCD and LCM The traffic lights at three different road crossings change after every 48 sec, 72 sec, and 108 sec respectively. If they all change simultaneously at 8:20:00hours, then at what time will they change again simultaneously? 
okay seriously how would someone even guess we have to do this by LCM GCD method ?
 A: It seems you need some help in understanding how $\,\rm lcm\,$ plays a role in such problems. Let's count time in seconds, relative to the starting time 8:20 when they all changed. Then light $\#1$ changes every $48$ seconds, i.e. at times $48n = 48, 96,\ldots =$ all multiples of $48,\,$ which are listed  in the first row in the table below. $ $ Similarly for lights $\#2$ and $\#3$ in the $2$nd and $3$rd rows.
$$\begin{array}{|r|r|rrrrrrrr|}
\hline
\#1  & 48n  & 48 \!&\!    \!&\! 96 \!&\!     \!&\! \color{#c00}{144}  \!&\! 192  \!&\!      \!&\! 240  \!&\! \color{#c00}{288}  \!&\!     \!&\! 336   \!&\!      \!&\! 384  \!&\! \color{#0a0}{432}  \!&\! \ldots\, \\
\hline
\#2  & 72n  &    \!&\! 72 \!&\!    \!&\!     \!&\! \color{#c00}{144}  \!&\!      \!&\! \color{#c00}{216}  \!&\!      \!&\! \color{#c00}{288}  \!&\!      \!&\!      \!&\! 360  \!&\!      \!&\! \color{#0a0}{432}  \!&\! \ldots\,\\
\hline
\#3 &  108n \!&\!    \!&\!    \!&\!    \!&\! 108 \!&\!      \!&\!      \!&\! \color{#c00}{216}  \!&\!      \!&\!      \!&\! 324  \!&\!      \!&\!      \!&\!      \!&\! \color{#0a0}{432}  \!&\! \ldots\,\\
\hline
\end{array}$$
The times when $2$ of $3$ lights change are shown in $\rm\color{#c00}{red},$ and $\rm\color{#0a0}{green}$ shows the time when all $3$ change.  By  construction, the number $\color{#0a0}{432}$ is a common multiple of $48, 72,108.\,$ Since no prior common multiple of all three occurs, it is their least common multiple $(\rm lcm).$
Recall, as I explained in your prior question, the $\rm lcm$ is characterized by the universal property
$${ 48,72,108\mid m\iff \overbrace{{\rm lcm}(48,72,108)}^{\large \color{#0a0}{432}}\mid m}$$
This says that the times $m$ when all $3$ change $ $ (i.e. when $m$ is a common multiple of $48,72,108),\,$ are equivalent to the times when $m$ is a multiple of their $\,{\rm lcm} = \color{#0a0}{432}.\,$ Interpreted in terms of the above table, this is true because the displayed pattern repeats if we consider the times $\!\!\pmod{\!\color{#0a0}{432}}$ 
A: here's an expansion on lulu's writing:


*

*8:20:00 all three are changing 

*8:20:48 light 1 changes  48 seconds

*8:21:12 light 2 changes  72 seconds

*8:21:36 light 1 changes  96 seconds

*8:21:48 light 3 changes  108 seconds

*8:22:24 light 1 and light 2 change  144 seconds

*8:23:12 light 1 changes   192 seconds

*8:23:36 light 2 and light 3 change  216 seconds

*8:24:00 light 1 changes  240 seconds

*8:24:48 light 1 and light 2 change  288 seconds

*8:25:24 light 3 changes  324 seconds

*8:25:36 light 1 changes  336 seconds

*8:26:00 light 2 changes  360 seconds

*8:26:24 light 1 changes  384 seconds

*???     all three change  ??? seconds 


this took me very much more time than before bill's first comment to work out by mind when at least with computer I can compute the final time fairly fast with LCM so maybe they'd think to use it because it took them so much time otherwise. 
A: "okay seriously how would someone even guess we have to do this by LCM GCD method ?"
If the first light blinks ever $48$ seconds then any time it blinks will be at a mmultiple of $48$.  So the answer when the all blink will have to be a multiple of $48$ because those are the only times the first one blinks.
The second only blinks at a multiple of $72$ seconds.  So the answer has to be a common multiple of both $48$ and $72$.
And so on.  The answer has to be a common multiple of all $48$,  $72$, and $108$.  What is a common multiple of $48$, $72$, and $108$.
Well... notice the first light will blink at: $48, 96, 144, 192, 240, 288, 336, 384, 432, 480,......$
The second light at: $72, 144, 216, 288, 360, 432, 504, 576, 648, 720 ....$
The third at: $108, 216, 324, 432, 540, 648, 756, 864, 972, 1080 ....$
What number do they all have in common?  Well.... I can see it is $432$.
Dang.  That was tedious.  Maybe there is an easier way to figure out what is the least common mulitple of $48, 72$ and $108$.
