James was trying to sleep one night but there was too much noise around him. His clock ticked every 20 seconds; a tap was dripping every 15 seconds and his pet dog snored every 27 seconds. Lid M
He noticed on his clock that all the three events happened together on the stroke of midnight.
(a) After how many minutes would all the three events happen together again?
(b) How many times would all the three events happen together again between midnight and one
o'clock?

It will occur zero times between midnight and one o'clock.

Step-by-step explanation:

Least Common Multiple (LCM)

Three events keep James from sleeping: his clock ticking every 20 seconds, a tap dripping every 15 seconds, and his dog snoring every 27 seconds.

All three events happened together at midnight. They will happen together again the first time the numbers 20, 15, and 27 have a common multiple. This is the LCM.

List the prime factors of each number:

20: 2,2,5

15: 3,5

27: 3,3,3

Now multiply all the factors the maximum number of times they appear:

LCM=2*2*3*3*3*5=540

(a) All the events will happen together again after 540 minutes.

(b) Since 540 minutes = 9 hours, this event won't happen again until 9 am. Thus, it will occur zero times between midnight and one o'clock.

a negative exponent is translated as follows:

so, we have