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 ?
On
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 $\!\!\pmod{\!\color{#0a0}{432}}$
On
"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 $108$.
here's an expansion on lulu's writing:
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.