Question

称一个数为“看似质数”，如果它是合数但不能被 $2, 3$ 和 $5$ 整除。最小的三个看似质数是 $49, 77$ 和 $91$ 。小于 $1000$ 的质数有 $168$ 个。小于 $1000$ 的看似质数有多少个？

Call a number "prime-looking" if it is composite but not divisible by $2, 3,$ or $5.$ 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 $1000?$

$100$

$102$

$104$

$106$

$108$

答案：A

知识点：容斥原理质数区间内整数计数

难度评级：1950

解答：

在 $1$ 到 $999$ 的 $999$ 个数中，由容斥原理，能被 $2, 3$ 或 $5$ 中至少一个整除的数共有 $\begin{aligned} &499 + 333 + 199 - 166 \\ &\quad {}- 99 - 66 + 33 = 733 \end{aligned}$

因此与 $2, 3, 5$ 都互质的数共有 $999 - 733 = 266$ 个；其中有 $165$ 个质数（ $168$ 个质数去掉 $2, 3, 5$ ），另有 $1$ 既不是质数也不是合数。

剩下 $266 - 165 - 1 = 100$ 个就是看似质数。

所以正确答案是 A。

Among the $999$ numbers from $1$ to $999,$ inclusion-exclusion gives $\begin{aligned} &499 + 333 + 199 - 166 \\ &\quad {}- 99 - 66 + 33 = 733 \end{aligned}$ that are divisible by $2, 3,$ or $5.$

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.

2005 AMC 12A 第 18 题

答案：A

难度评级：1950

解答：

其他年份的第 18 题