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The slow wall clock

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A wall clock with a $12$ -hour dial is set exactly right at the moment $0$ . From then on it runs steadily slow, losing exactly $2$ minutes of every real hour. Over the next $100$ days of real time, at how many moments does the dial show the correct time of day? (Count moments up to and including the end of day $100$ , and do not count the moment $0$ itself.)

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Hint

The dial is correct whenever the total time lost is a whole number of full dial turns.

Solution

After $t$ real minutes the clock has lost $\frac{t}{30}$ minutes, because it loses $2$ minutes per $60$ .
A $12$ -hour dial shows the right time exactly when the lost time is a multiple of $720$ minutes, so $t$ must be a multiple of $720 \times 30 = 21600$ minutes, which is $15$ days.
The first $100$ days contain $\frac{100}{15} \approx 6.67$ such periods, so the moments are at days $15$ , $30$ , $45$ , $60$ , $75$ and $90$ .
The number of moments is $6$ .

Answer: $6$

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