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Spotlights over a stage

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A stage has $50$ spotlights numbered $1$ to $50$ , all off. A lighting crew makes $3$ passes: in pass $1$ they toggle every spotlight, in pass $2$ every spotlight whose number is even, and in pass $3$ every spotlight whose number is a multiple of $3$ . Toggling turns a spotlight on if it is off and off if it is on. How many spotlights are on after the $3$ passes?

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Hint

A spotlight is on when it was toggled $1$ or $3$ times.

Solution

Every spotlight is toggled in pass $1$ , so it ends on exactly when passes $2$ and $3$ together toggle it an even number of times.
That happens when the number is divisible by both $2$ and $3$ , or by neither.
Up to $50$ there are $8$ multiples of $6$ , and $50 - 25 - 16 + 8 = 17$ numbers divisible by neither.
So $8 + 17 = 25$ spotlights are on, and the answer is $25$ .

Answer: $25$

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