How often between 11 O'clock and 12 O'clock are the hands of a clock in integral number of minutes apart?

As per the given above question , we can say that
At 11 O'clock, the hour hand is 4 spaces apart from the minute hand. Since there are 60 spaces in one hour, so (60 - 4) times, i.e., 56 times the hands of the clock are an integral number of minutes apart.
Hence , required answer is 56 times .