1990 AMC 12 第 11 题

Question

小于

50

的正整数中，有多少个数的正因数个数为奇数？

How many positive integers less than

50

have an odd number of positive integer divisors?

3

5

7

9

11

因数通常成对出现，即

d

与

\frac{n}{d}

Divisors normally pair as

d

and

\frac{n}{d}

只有完全平方数会出现未配对的因数

An unpaired divisor occurs exactly for a perfect square

一个正整数的因数个数为奇数，当且仅当它是完全平方数。小于

50

的完全平方数为

1,4,9,16,25,36,49\text{，}

所以共有

7

个。

所以正确答案是 C。

A positive integer has an odd number of divisors exactly when it is a perfect square. The squares below

50

are

1,4,9,16,25,36,49,

so there are

7.

Thus the correct answer is C.

其他年份的第 11 题