Question

The traffic lights at three different road crossings change after every 48 seconds 72 seconds and 18 18 seconds respectively if they change symmetry asli at 8:00
a.M. At what time will they change together again

Answer 1

Step 1 :-

Find the prime factors:-

$48 = {2}^{4}\times3\\72 = {2}^{3}\times{3}^{2}\\18 = 2\times {3}^{2}$

Step 2 :-

Find the LCM :-

LCM

$= {2}^{4} \times {3}^{2} \\ = (2 \times 2 \times 2 \times 2) \times (3 \times 3) \\ = 16 \times 9 \\ = 144 \: seconds$

Step 3 :-

Convert seconds into minutes:-

$144 \: sec = \frac{144}{60} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 2.4 \: min$

Step 4 :-

Find the time it will change together again :-

(8 a.m) + (2.4 min) = 8.02.04 a.m

Therefore,

They will change together at 8.02.04 a.m.

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Answer 2

$\large \green{ \fcolorbox{gray}{black}{ ☑ \: \textbf{Verified \: answer}}}$

If the traffic lights change simultaneously at 8 a.m, then they will change simultaneously again after the LCM of the duration i.e. 48 s, 72 s and 108 s.

LCM of these durations by prime factorisation

• 48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3
• 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
• 108 = 2 × 2 × 3 × 3 × 3 = 2² × 3³

LCM of 48, 72 and 108 is 2⁴ × 3³ = 432 seconds.

Hence, They will change after 432 seconds i.e. 7 minutes 12 seconds.

The traffic lights will change after:

8 am + 7 minutes 12 seconds

08 : 07 : 12 am