Question

Three friends set their clock alarms in such a way that they ring after every 12, 15 and 18
minutes respectively. If initially all the alarms ring at the same time, then find after how
many minutes will they ring together again.

Answer 1

━━━━━━━━━━━━━━━━━━━━

✺Answer:

♦️GiveN:

• Three Clock alarms ring after 12, 15 and 18 minutes respectively.

♦️To FinD:

• After what time they will rang together...?

━━━━━━━━━━━━━━━━━━━━

✺Explanation of Q.

It is given that each of the 3 clocks rang after a certain interval of time which is different for all. These time intervals are 12, 15 and 18 minutes respectively. This means:

♣️1st clock rang after multiples of 12 from initial time. 2nd clock rang after multiples of 15 from initial time.3rd clock rang after multiples of 18 from initial time.

We have to find a time when all three will ring together, so that would the time which come common in the multiple table of 12 mins, 15 mins and 18 mins. This would the least multiple of 12, 15 and 18.

They will ring at LCM of 12, 15 and 18

━━━━━━━━━━━━━━━━━━━━

✺Solution:

Finding the LCM of 12, 15 and 18

$\large{ \rm{ \rightarrow \: 12 = \green{ \underline{2}} \times 2 \times \red{ \underline{ 3}}}} \\ \\ \large{ \rm{ \rightarrow \: 15 = \red{ \underline{3}} \times 5}} \\ \\ \large{ \rm{ \rightarrow \: 18 = \green{ \underline{2 }}\times \red{ \underline{3 }}\times 3}} \\ \\ \large{ \rm{ \purple{lcm} = \green{\underline{2}} \times \red{\underline{3}} \times 2 \times 3 \times 5 = \boxed{180}}}$

⏺ So, the Three clocks alarms will ring after 180 minutes.

━━━━━━━━━━━━━━━━━━━━