At four o'clock the angle between the minute and the hour hand is 120o. This angle reduces at the rate of 5.5o per minute till it reaches zero and thereafter the angle increases at the same rate. The two hands are at right angles when the angle between them changes by 30o and 210o (i.e., 120o - 90o and 120o + 90o).
The angle changes by 5.5o in 1 minute.
So, the angle changes by 30o in 15.5×30o=5511 minutes.
The angle changes by 210o in 15.5×210o=38211 minutes.
Thus, the hands are at right angle at 5511 past 4 and again at 38211 minutes past 4.