Question
The traffic lights at three different road crossing change after every 48 secons, 72 seconds and 108 seconds respectively. If they all change simultaneously at 8 a.m. then at what time will they again change simultaneously?
✓
Answer
Let us find the LCM of 48, 72 and 108 through prime factorisation:
$\begin{array}{c|c} 2 & 48 \\ \hline 2 & 24\\ \hline2&12\\ \hline2&6\\ \hline&3 \end{array}$ $\begin{array}{c|c} 2 & 72 \\ \hline 2 & 36\\ \hline2&18\\ \hline3&9\\ \hline&3 \end{array}$ $\begin{array}{c|c} 2 & 108 \\ \hline 2 & 54\\ \hline3&27\\ \hline3&9\\ \hline&3 \end{array}$
$48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3$
$72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$
$108 = 2 \times 2 \times 3 \times 3 \times 3 = 2^2 \times 3^3$
LCM of $48,72,108$ is $2^4 \times 3^3$
= 16 × 27sec
= 432sec
=7min 12sec
Three bells toll together after 7min 12sec.
$\begin{array}{c|c} 2 & 48 \\ \hline 2 & 24\\ \hline2&12\\ \hline2&6\\ \hline&3 \end{array}$ $\begin{array}{c|c} 2 & 72 \\ \hline 2 & 36\\ \hline2&18\\ \hline3&9\\ \hline&3 \end{array}$ $\begin{array}{c|c} 2 & 108 \\ \hline 2 & 54\\ \hline3&27\\ \hline3&9\\ \hline&3 \end{array}$
$48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3$
$72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$
$108 = 2 \times 2 \times 3 \times 3 \times 3 = 2^2 \times 3^3$
LCM of $48,72,108$ is $2^4 \times 3^3$
= 16 × 27sec
= 432sec
=7min 12sec
Three bells toll together after 7min 12sec.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
From a point $P$ on the ground the angle of elevation of the top of a tower is $30^{\circ}$ and that of the top of a flag staff fixed on the top of the tower, is $60^{\circ}$. If the length of the flag staff is 5 m , find the height of the tower.
→The dimensions of a room are 14m × 10m × 6.5m. There are two doors and 4 windows in the room. Each door measures 2.5m × 1.2m and each window measures 1.5m × 1m. Find the cost of painting the four walls of the room at ₹ 35 per $m^2.$
→Find an AP whose $4^{\text {th }}$ term is 9 and the sum of its $6^{\text {th }}$ and $13^{\text {th }}$ terms is 40 .
→The inside perimeter of a running track (shown in the following figure) is 400m. The length of each of the straight portion is 90m and the ends are semi-circles. If the track is everywhere 14m wide. find the area of the track. Also find the length of the outer running track.
A footpath of uniform width runs all around the inside of a rectangular field 54m long and 35m wide. If the area of the path is $420m^2$., find the width of the path.
→If $\alpha,\ \beta,\ \gamma$ are the zeroes of the polynomial $p(x) = 6x^3 + 3x^2^- 5x + 1$, find the value of $\Big(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\Big).$
→An electronic device makes a beep after every 60 seconds. Another device makes a beep after 62 seconds. They beeped together at 10 a.m. At what time will they beep together at the earliest?
Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.
If $\cos\theta=\frac{3}{5},$ find the value of $\frac{\sin\theta-\frac{1}{\tan\theta}}{2\tan\theta}.$
→If $\text{cosec}\theta+\cot\theta=\text{p},$ then prove that $\cos\theta=\frac{\text{p}^2-1}{\text{p}^2+1}.$
→