x is a positive integer and it is a product of p distinct prime number

Question

x is a positive integer and it is a product of p distinct prime numbers. What is the value of p?

(1) x is divisible by exactly 8 positive integers.

(2) x and ﻿  ﻿2*3^3  have the same numbers of divisors.

chetan2u · Answer

Surprisingly, 86% have answered it D as per the given OA, which I believe is wrong.

(1) x is divisible by exactly 8 positive integers.

(2) x and ﻿  ﻿2*3^3  have the same numbers of divisors.

Both the statements give the same information that is, the number of positive divisors are 8. 
8=(1+7), so the number could be  a^7 , that is only one prime factor. 
8=2*4=(1+1)(3+1), so the number could be  a^1*b^3 , that is two prime factors. 
8=2*2*2=(1+1)(1+1)(1+1), so the number could be  aabc , that is three prime factors.

Thus p can be 1, 2 or 3.

Insufficient

E

A poor quality question unless I have missed something.

ktzsikka · Answer

Hi  chetan2u 
I think, you missed " distinct ".

ktzsikka · Answer

Here is the  OE

Solution:
Solution
 Step 1: Analyse Question Stem 
x is a positive integer.
x is a product of p distinct prime numbers.
 ﻿x=P_1*P_2*P_3…P_p=P  ﻿ where P is a prime number.
We need to find the value of p.

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
Statement 1: x is divisible by exactly 8 positive integers.

Since, x can be written as ﻿ P_1*P_2*P_3…P_p  
Number of factors of x =﻿(1+1)*(1+1)*……..(1+1) \ upto \ p \ times=2^p(1+1)∗(1+1)∗……..(1+1) upto p times= 2^p 
 ﻿

As per the given statement:

Number of factors of x = 8
Thus,﻿  2^p=8= 2^3

﻿ ⟹ p = 3
Hence, statement 1 is sufficient and we can eliminate answer options B, C and E

Statement 2: : x and ﻿ 2* 3^3∗3  has same numbers of divisors.

Number of divisors of x = number of divisors of  ﻿2* 3^3   =  (1+1) *(3+1) =8 
 ﻿2^p=8= 2^3  
 ﻿ 
Thus, p = 3
Hence, statement 2 is also sufficient and the correct answer is Option D.

chetan2u · Answer

If it is 2^3, there is only one distinct prime number and that is 2.
How can p be 3.

It is a poor quality question with equally poor quality of solution.

ktzsikka · Answer

Okay.. Now I understood your PoV.
Yes, 2x2x2 is also a product of (1) distinct prime number. But I think the question maker must have thought that the factors should not be repetitive. Maybe " different " should also be added to the question, to clear the ambiguity.
And yeah, OE justifies the question, so it's also incorrect.

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x is a positive integer and it is a product of p distinct prime number

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