AMC8 2012

AMC8 2012 · Q18

AMC8 2012 · Q18. It mainly tests Primes & prime factorization, Parity (odd/even).

What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?

最小的既非质数又非平方数，且没有小于50的质因数的正整数是多少？

(A) 3127 3127

(B) 3133 3133

(D) 3139 3139

(E) 3149 3149

Answer

Correct choice: (A)

正确答案：（A）

Solution

The problem states that the answer cannot be a perfect square or have prime factors less than $50$. Therefore, the answer will be the product of at least two different primes greater than $50$. The two smallest primes greater than $50$ are $53$ and $59$. Multiplying these two primes, we obtain the number $3127$, which is also the smallest number on the list of answer choices. So we are done, and the answer is $\boxed{\textbf{(A)}\ 3127}$.

题目要求答案不能是完全平方数，也不能有小于 $50$ 的质因数。因此，答案至少是两个不同大于50的质数的乘积。大于50的最小两个质数是 $53$ 和 $59$。它们的乘积是 $3127$，这也是选项中最小的数。因此答案是 $\boxed{\textbf{(A)}\ 3127}$。

Topics