How many of the smallest 200 positive integers are divisible by 3,5, and/or 7?

  108

Step-by-step explanation:

The number that are divisible by 3 and 5 and 7 is the number of integer times 3·5·7 = 105 divides 200. That is 1.

The number that are divisible by 3 and 5 is floor(200/15) = 13, so 12 additional numbers.

The number that are divisible by 3 and 7 is floor(200/21) = 9, so 8 additional numbers.

The number that are divisible by 5 and 7 is floor(200/35) = 5, so 4 additional numbers.

Of the floor(200/3) = 66 numbers divisible by 3, 21 of them are also divisible by 5 or 7 or both, so 66 -21 = 45 additional numbers.

Of the floor(200/5) = 40 numbers divisible by 5, 17 of them are also divisible by 3 or 7 or both, so 40 -17 = 23 additional numbers.

Of the floor(200/7) = 28 numbers divisible by 7, 13 of them are also divisible by 3 or 5 or both, so 28 -13 = 15 additional numbers.

The total number of numbers divisible by 3, 5, or 7 in the first 200 positive integers is ...

  1 +12 +8 +4 +45 +23 +15 = 108

108 of the smallest 200 positive integers are divisible by 3, 5, and/or 7.

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step-by-step explanation: