2005 AMC 12A Problem 18

Call a number “prime-looking” if it is composite but not divisible by

2,

3,

5.

The three smallest prime-looking numbers are

49,

77,

and

91.

There are

168

prime numbers less than

1000.

How many prime-looking numbers are there less than

1000?

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Solution

Among the

999

numbers from

1

999,

inclusion-exclusion gives

\begin{aligned} &499 + 333 + 199 - 166 \\ &\quad {}- 99 - 66 + 33 = 733 \end{aligned}

that are divisible by

2, 3,

5.

That leaves

999 - 733 = 266

numbers coprime to

2, 3, 5.

Of these,

165

are primes (the

168

primes minus

2, 3, 5

), and

1

is neither prime nor composite. The remaining

266 - 165 - 1 = 100

numbers are prime-looking. Thus, the correct answer is A.