Explanation:

- The three alarm clocks ring at regular intervals: every 20, 21, and T minutes.

- They all beep together at 12 noon. To find when they'll next beep together, find the least common multiple (LCM) of the intervals.

- The LCM of 20 and 21 is 420 minutes.

- To find when all three clocks beep together, calculate the LCM of 420 and T.

- We need the smallest value of T from the options, where LCM(420, T) is the smallest.

Option 1: T = 8 minutes

- LCM(420, 8) = 840

Option 2: T = 18 minutes

- LCM(420, 18) = 1260

Option 3: T = 28 minutes

- LCM(420, 28) = 420 (since 28 divides 420)

Option 4: T = 38 minutes

- LCM(420, 38) = 7980

- Answer: Option 3 - 28 minutes is correct.