When do the smoke alarm and air conditioner beep at the same time ?

Question

Smoke alarm beeps every 60 seconds

air conditioner beeps every 72 seconds

Answer 1

Two facts determine the answer:

1)     The simultaneous beeps will occur whenever there is a 60 second time differential or a 72 second time differential.

2)     There is a 12 second difference between the two intervals.

That means that the first simultaneous beep pair will occur at the 5th 72 second interval, which is 360 seconds from when both are turned on. That means the 6^th 60 second interval for the smoke alarm. At that point, the count begins anew. The 72 second differential is therefore, never reached, so the simultaneous beeps will occur every 360 seconds.

Mathematically,

60a = 72b, where both a and b are integers

a = 1.2*b

Since a (and b) is an integer, 1.2*b must be an integer, which only happens every five 72 second intervals, i.e. b = 5, 10, 15, … , or every 5 72-second intervals, or every 360 seconds.