How many times are an hour hand and a minute hand of a clock at right angles during their motion from 1.00 p.m. to 10.00 p.m.?

The hands are at right angles when |30H - 5.5M| = 90 or 270. Measuring t in minutes after 12:00, right angles occur at t = (90 + 180k)/5.5. Between 1:00 p.m. (t = 60) and 10:00 p.m. (t = 600) this holds for k = 2 to 17, i.e. 16 times. 3:00 and 9:00 are each counted once, so the naive 'two per hour x 9 hours = 18' overcounts.