Question

QJ. Train1 whistles every 120 minutes, Train2 whistles every 140 minutes, and Train3 whistles every 160 minutes. If they all whistle at 3PM on the same day, at what time will they next all whistle together?​

Answer 1

$\large\underline{\sf{Solution-}}$

★ Given that,

• Train1 whistles every 120 minutes,

• Train2 whistles every 140 minutes,

• Train3 whistles every 160 minutes.

So, in order to find that they will whistle together after 3 PM again at the time equals to LCM of 120, 140, 160

★ So, lets find the LCM using prime factorization method.

$\red{\bf :\longmapsto\:Prime \: factorization \: of \: 120}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:120 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:60 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:30\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:15 \:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

$\red{\rm :\longmapsto\:Prime \: factorization \: of \: 120 = {2}^{3} \times 3 \times 5}$

$\blue{\bf :\longmapsto\:Prime \: factorization \: of \: 140}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:140 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:70 \:\:}} \\\underline{\sf{5}}&\underline{\sf{\:\:35\:\:}} \\ {\underline{\sf{7}}}& \underline{\sf{\:\:7 \:\:}} \\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

$\blue{\rm :\longmapsto\:Prime \: factorization \: of \: 140 = {2}^{2} \times 5 \times 7}$

$\green{\bf :\longmapsto\:Prime \: factorization \: of \: 160}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:160 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:80 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:40\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:20 \:\:}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:10\:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

$\green{\rm :\longmapsto\:Prime \: factorization \: of \: 160 = {2}^{5} \times 5}$

Hence,

$\bf :\longmapsto\:LCM(120, 140, 160) = {2}^{5} \times 3 \times 5 \times 7$

$\bf :\longmapsto\:LCM(120, 140, 160) = 3360 \: minutes.$

★ It implies, the trains whistles together after 3360 minutes, that is after 56 hours, they whistle together.