How many prime factors does the number X have?

Question

e-GMAT Question:

1) X is divisible by 53
2) X is divisible by 4 distinct integers.

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are NOT sufficient

This is  Question 8 of  The e-GMAT Number Properties Marathon

Go to   Question 9 of the Marathon

Hatakekakashi · Answer

if x is divisible by 53, it can have other primes as well

insufficient

x is divisible by 4 distinct integers

1, x , and 2 others

now here the other 2 must be prime

14 = 1 2 7 14
15 = 1 3 5 15 
sufficient

but if

x = cube of a prime i.e 2^3 , which has 4 factors

we get 1 prime

hence insufficient

combining x can be cube of a prime or multiple of 2 squares of prime

(E) imo

amanvermagmat · Answer

Statement 1

If X is divisible by 53, then X could just be '53' (in which case only 1 prime factor, 53) or X could be '53*2 = 106' (in which case 2 prime factors, 2 and 53). 
So we cannot be sure about the number of prime factors of X.  Not Sufficient .

Statement 2

X is divisible by 4 distinct integers, this means X has total 4 factors. Now any number having 4 factors in total, could have two kinds of prime factorisation:

a) X = p^3. So X could be cube of a prime number. In this case X will have total 4 factors, but only 1 prime factor (p). 
b) X = p1 * p2. So X could be a product of two distinct prime numbers. In this case also X will have total 4 factors, but there are 2 prime factors (p1 and p2).

So we cannot be sure about number of prime factors of X.  Not Sufficient .

Combining the statements , we know X is divisible by 53 and X has 4 total factors.

So if X looks like p^3, then X = 53^3. In this case X has only 1 prime factor 53. 
But if X looks like p1 * p2, then X = 53*p2 (product of 53 with any other prime number). In this case X has 2 prime factors 53 and p2.

So we still cannot be sure, as there are two possibilities.  Not Sufficient .

Hence  E answer

EgmatQuantExpert · Answer

Solution:

Step 1: Analyse Statement 1: 
  X  is divisible by  53 . 
 • The given statement tells us that  X  is a multiple of  53 .
 o So, from this statement we can write  X  in the form of  53k , where  k   is any positive integer.  
• To find the number of prime factors of  X=53k , we need to have information on the value of  k .  
As we do not have any information on  k , 
Statement 1 alone is NOT sufficient to answer the question. 
Hence, we can eliminate answer choices A and D. 
 Step 2: Analyse Statement 2: 
 X  is divisible by  4  distinct integers
 • The given statement tells us that  X  has  4  factors. 
 o It can be of form  P1 * P2 , where  P1 and  P2  are the prime factors of the number. 
o Or, it can be of the form  P1^3 , where  P1  is the prime factor of the number.  
• Hence, it can have either  1 or  2  prime factors.  
Statement 2 alone is NOT sufficient to answer the question. 
 Step 3: Combine both Statements: 
 • From the first statement, we got  X=53k 
• From the second statement we got  X has  4  factors
 o Combining the statements, we may encounter two possible cases: 
  It can be of form  53 * P2 , where  P2  is a prime factor of the number. 
 Or, it can be of form 53^3 .   
• Hence, it can have either  1  or  2  prime factors.

By combining both statements we did not get a unique answer.
Correct Answer: Option E

medic19 · Answer

Hi - wondering why is this a 700 question? Don't think there is anything out of the ordinary one needs to spot? Or is that on the basis of the responses given?

ailyninja · Answer

if x = cube of prime, does x have 4 DISTINCT integers??? I think the answer should be B.

exc4libur · Answer

1) X is divisible by 53: insufic.
if X=53y and y=1 then  pf(X)=53^1=(1+1)=2 
if X=53y and y=2 then  pf(X)=53^1•2^1=(1+1)(1+1)=4

2) X is divisible by 4 distinct integers: insufic.
if factors(X)={1,2,3,5} then  pf(X)=2^1•3^1•5^1=(2)(2)(2)=8 
if factors(X)={2,3,5,7} then  pf(X)=2^1•3^1•5^1•7^1=(2)(2)(2)(2)=16

(1&2) insufic.
if factors(X)={1,2,3,53} then  pf(X)=2^1•3^1•53^1=(2)(2)(2)=8 
if factors(X)={2,3,7,53} then  pf(X)=2^1•3^1•7^1•53^1=(2)(2)(2)(2)=16

Answer (E)

Fdambro294 · Answer

Statement 1:

If X is Divisible by 53, X can take the following Forms

Case 1: X = (53)'1st * (2)'1st ------ X contains 2 Prime Factors

or

Case 2: X = (53)'3rd ------ X contains 1 Unique Prime Factors

S1 NOT Suff.

Statement 2:

If X has 4 Distinct Factors, then X can take the following Forms (in which Variables A and B are Prime Bases)

Case 1: X = (A)^1st * (B)^1st ----- Total Factors = (1 + 1) * (1+ 1) = 4

if A = 53 and B = 2, then X has 2 Prime Factors

OR

Case 2: X = (A)^3rd ----- Total Factors = (1 +3) = 4

if A = 53, then X has 1 Unique Prime Factor

Statement 2 NOT Suff.

TOGETHER:

Since the SAME 2 Cases work for both Statements to prove Insufficiency, then the Statements will NOT be Sufficient Together.

Answer -E-

Sri07 · Answer

Easiest Approach Ever:

How many prime factors does the number X have?

1) X is divisible by 53
 53 is prime so however, x could be any number and have any number of prime factors. 
   So 1 itself is Insufficient

2) X is divisible by 4 distinct integers.
 X is divisible by 4 distinct integers. (Note: Not Prime factors) Well, this could be basically any number .. So in itself  statement 2 is Insufficient
 
1&2 combined also will not give us any new info. 
Like the number can be 53*(any number which has 4 integers as 
So IMO E

How many prime factors does the number X have?

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