This is a classic least common multiple (LCM) problem! Here's how to figure out when all the devices will beep together:

1. Find the Least Common Multiple (LCM):

* The LCM is the smallest number that is a multiple of all three numbers (8, 10, and 12).

* You can find the LCM using prime factorization or by listing multiples:

* Prime Factorization:

* 8 = 2 x 2 x 2

* 10 = 2 x 5

* 12 = 2 x 2 x 3

* LCM = 2 x 2 x 2 x 3 x 5 = 120

* Listing Multiples:

* Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112...

* Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120...

* Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120...

2. The Answer:

All three devices will beep together after 120 minutes, which is 2 hours.