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ktzsikka
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x is a positive integer and it is a product of p distinct prime number Updated on: 02 Dec 2021, 07:37
Originally posted by ktzsikka on 20 Oct 2021, 11:54.
Last edited by ktzsikka on 02 Dec 2021, 07:37, edited 1 time in total.
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Difficulty:

85% (hard)

Question Stats:

24% (01:36) correct 76% (01:37) wrong based on 46 sessions

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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.

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chetan2u
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x is a positive integer and it is a product of p distinct prime number 21 Oct 2021, 08:29
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ktzsikka
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.
Show more

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

Quote:
If x=\(a^p*b^q*c^r*d^s…\), then the number of positive divisors of x are \((p+1)(q+1)(r+1)(s+1)..\).
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(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.
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x is a positive integer and it is a product of p distinct prime number Updated on: 29 Nov 2021, 09:57
Originally posted by ktzsikka on 29 Nov 2021, 09:50.
Last edited by ktzsikka on 29 Nov 2021, 09:57, edited 1 time in total.
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chetan2u
ktzsikka
x is a positive integer and it is a product of p distinct prime numbers. What is the value of p?

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.

A poor quality question unless I have missed something.
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Hi chetan2u
I think, you missed "distinct".
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x is a positive integer and it is a product of p distinct prime number 29 Nov 2021, 09:56
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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.
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x is a positive integer and it is a product of p distinct prime number 29 Nov 2021, 19:28
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ktzsikka
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.
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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.
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x is a positive integer and it is a product of p distinct prime number 29 Nov 2021, 20:01
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chetan2u
ktzsikka
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.

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.
Show more

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.

Posted from my mobile device

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