Red light : every 4 secs
Blue light : every 5 secs
Purple light : every 6 secs

The idea is to find the lowest common multiple of the 3 numbers.

4 = 2 x 2
5 = 5 x 1
6 = 2 x 3

Only 5 has the factor 5.
4 and 6 both have the factor 2.
Only 6 has the factor 3.
Only 4 has two 2s.

So make all three numbers have one 3, one 5, and two 2s.

4 → x 5 x 3 = x 15 = 60
5 → x 2 x 3 x 3 = x 12 = 60
6 → x 5 x 2 = x 10 = 60

So 60 secs is the lowest common multiple.

This means that every 60 secs, the lights will flash together.

And 60 seconds = 1 min

The very next time the lights will flash together →9pm + 1 min
= 9.01pm

Ratio way to see it :

R B P
4 : 5 : 6
×15 ×12 ×10
↓

60 60 60