Question

The traffic lights at three different road crossings change after
every 48 seconds, 72 seconds and 108 seconds respectively. If
the change simultaneously at 8 a.m at what time time will they
change simultaneously again?​​

Answer 1

$\huge\boxed{\underline{\sf{\red{ᴀ}\green{ɴ}\pink{s}\orange{ᴡ}\blue{ᴇ}\pink{ʀ}}}}$

Required 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.

Then, finding the 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 is the product of highest degrees of factors. Then, 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. (Answer)

Answer 2

$\boxed{ \bf \: Required \: \: Answer:-}$

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.

Then, finding the 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 is the product of highest degrees of factors. Then, LCM of 48, 72 and 108 is 2⁴ × 3³ = 432 seconds.

$\orange{ \boxed{ \bf \: Hence:-}}$

They will change after 432 seconds i.e. 7 minutes 12 seconds. The traffic lights will change after:

$\sf \: ➙ \: \: 8 am + 7 \: minutes \: \: 12 \: seconds$

$\sf➙ \: 08 : 07 : \: 12 \: am \: is \: your \: answer$