1991 AMC 12 第 14 题

Question

若

x

是某个正整数的立方，且

d

是

x

的正因数个数，则

d

可能等于

If

x

is the cube of a positive integer and

d

is the number of positive integers that are divisors of

x,

then

d

could be

200

201

202

203

204

立方数的质因数分解中，每个指数都能被

3

整除

In the prime factorization of a cube, every exponent is divisible by

3

尝试只含一个质因数的立方数

Try a cube having only one prime factor

对任意质数

p

，取

x=p^{201}=(p^{67})^3

。这是一个立方数，它的正因数为

1,p,p^2,\ldots,p^{201}

，共有

202

个。因此

202

可以出现。

因此正确答案为 C。

For any prime

p,

take

x=p^{201}=(p^{67})^3.

This is a cube, and its positive divisors are

1,p,p^2,\ldots,p^{201},

a total of

202.

Thus

202

can occur.

Thus the correct answer is C.

其他年份的第 14 题