If x is an integer that has exactly three positive divisors (these include 1 and x), how many positive divisors does x^3 have?
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?
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?
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
itself. So, x
can be 4, 9, 25, ... For example divisors of 4 are: 1, 2=prime, and 4 itself.
, so it has 6+1=7 factors (check below for that formula).
Else you can just plug some possible values for x
: say x=4
--> # of factors of 2^6 is 6+1=7.
Answer: D.Finding the Number of Factors of an Integer
First make prime factorization of an integer n=a^p*b^q*c^r
, where a
, and c
are prime factors of n
, and r
are their powers.
The number of factors of n
will be expressed by the formula (p+1)(q+1)(r+1)
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
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.
NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!
PLEASE READ AND FOLLOW: 11 Rules for Posting!!!
RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;
COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat
DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests?
25 extra-hard Quant Tests