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Right angles at whole minutes

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An analog clock has hands that move continuously: the hour hand turns $0.5$ degrees per minute and the minute hand $6$ degrees per minute, and both point straight up at $0 : 00$ . Consider the $1440$ whole-minute times from $0 : 00$ to $23 : 59$ in one day. At how many of them do the hands form an angle of exactly $90$ degrees?

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Hint

Write the angle between the hands after $m$ minutes as $5.5 m$ degrees modulo $360$ , and find which whole $m$ give $90$ or $270$ .

Solution

After $m$ minutes the minute hand is ahead of the hour hand by $5.5 m$ degrees modulo $360$ , so a right angle needs $5.5 m$ to be $90$ or $270$ modulo $360$ .
Multiplying by $2$ turns this into $11 m$ being $180$ or $540$ modulo $720$ .
Since $11$ and $720$ share no factor, each of these has exactly one solution $m$ in a $12$ -hour stretch, namely $540$ and $180$ minutes, which are $9 : 00$ and $3 : 00$ .
A day holds $2$ such stretches, so the count is $2 \times 2 = 4$ .

Answer: $4$

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