nades09 wrote:

If x is an integer that has exactly three positive divisors (these include 1 and x), how many positive divisors does x^3 have?

A. 4

B. 5

C. 6

D. 7

E. 8

I am not sure how the answer is derived here

If x=2, then X^3 =8 and 8 has 4 divisors - 1,2,4,8

But if x=9, then 9^3 =3^6, will have 7 divisors. So isn't the number of positive divisors dependent on the value of x?

Please explain

Thanks

x cannot be 2, because 2 has only two divisors 1 and 2, not three as given in the stem.

If x is an integer that has exactly three positive divisors (these include 1 and x), how many positive divisors does x^3 have?A. 4

B. 5

C. 6

D. 7

E. 8

Important property: the

number of distinct factors of a perfect square is ALWAYS ODD.

The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square. (A perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is a perfect square).

Hence, since given that x has 3 (odd) divisors then

x is a perfect square, specifically square of a prime. The divisor of

x are:

1,

\sqrt{x}=prime and

x itself. So,

x can be 4, 9, 25, ... For example divisors of 4 are: 1, 2=prime, and 4 itself.

Now,

x^3=(\sqrt{x})^6=prime^6, so it has 6+1=7 factors (check below for that formula).

Answer: D.

Else you can just plug some possible values for

x: say

x=4 then

x^3=64=2^6 --> # of factors of 2^6 is 6+1=7.

Answer: D.

Finding the Number of Factors of an IntegerFirst make prime factorization of an integer

n=a^p*b^q*c^r, where

a,

b, and

c are prime factors of

n and

p,

q, and

r are their powers.

The number of factors of

n will be expressed by the formula

(p+1)(q+1)(r+1).

NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450:

450=2^1*3^2*5^2Total number of factors of 450 including 1 and 450 itself is

(1+1)*(2+1)*(2+1)=2*3*3=18 factors.

So, the # of factors of x=a^2*b^3, where a and b are different prime numbers is (2+1)(3+1)=12.

Hope it's clear.

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