AMC 8 Daily Practice Round 9

Complete problem set with solutions and individual problem pages

Problem 16 Medium

Among the 100 natural numbers from 1 to 100, how many numbers are not multiples of 2, not multiples of 3, and not multiples of 5?

  • A.

    24

  • B.

    25

  • C.

    26

  • D.

    27

  • E.

    28

Answer:C

Among the 100 natural numbers from 1 to 100:

- Multiples of 2: 50

- Multiples of 3: 33

- Multiples of 5: 20

- Multiples of both 2 and 3: 16

- Multiples of both 2 and 5: 10

- Multiples of both 3 and 5: 6

- Multiples of 2, 3, and 5: 3

Using the principle of Inclusion-Exclusion, the total count of numbers that are multiples of 2, 3, or 5 is:

50 + 33 + 20 - 16 - 10 - 6 + 3 = 74

Thus, the count of numbers not being multiples of 2, 3, or 5 is:

100 - 74 = 26

The answer is \text{C}.

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