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: \(450=2^1*3^2*5^2\)
Total 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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