How many multiples of 15 are there between -200 and 200?

Correct.
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For example, 200 is divisible by 5 but not by 3 since the sum of its digits ($$2+0+0$$) is not divisible by 3. The next closest multiple of 5 is 195, which satisfies the conditions for 3 and 5 and is thus divisible by 15. The least multiple is thus -195, and the greatest is 195.
To count multiples, subtract the extremes, divide by 15, and don't forget to add 1:
>$$\displaystyle \frac{195-(-195)}{15} + 1 = \frac{390}{15} + 1 = 27$$.

Incorrect.
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25

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27

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29