24 hour analog clock problem.

Question

Find all instances on a 24 hour analog clock where if the hour and minute hands are interchanged we get a valid time.

Solution:

Let the position of the hour and minute hands be measured clockwise in degrees from the vertical.

Let:

    X  =  position of hour hand

  24X  =  position of minute hand  (mod 360)  

After interchanging the hour and minute hands:

  24X  =  position of hour hand    (mod 360) 

 576X  =  position of minute hand  (mod 360)  =  X 

We seek all solutions to:

     X  =  576X (mod 360)

     0  =  575X (mod 360)

     X  =  (360/575)N; N  =  0, 1, 2, …, 574