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M01-35

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M01-35  [#permalink]

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New post 15 Sep 2014, 23:16
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A
B
C
D
E

Difficulty:

  35% (medium)

Question Stats:

69% (00:47) correct 31% (01:02) wrong based on 178 sessions

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Re M01-35  [#permalink]

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New post 15 Sep 2014, 23:16
Official Solution:

If \(@x\) is the number of distinct positive divisors of \(x\), what is the value of \(@(@90)\)?

A. 3
B. 4
C. 5
D. 6
E. 7


Finding the Number of Factors of an Integer

First 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.

Back to the original question:

The question defines \(@x\) as the number of distinct positive divisors of \(x\). Say \(@6=4\), as 6 have 4 distinct positive divisors: 1, 2, 3, 6.

Question: \(@(@90)=\)?

\(90=2*3^2*5\), which means that the number of factors of 90 is: \((1+1)(2+1)(1+1)=12\). So \(@90=12\). Next, \(@(@90)=@12\). Now, since \(12=2^2*3\), then the number of factors of 12 is: \((2+1)(1+1)=6\).


Answer: D
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Re: M01-35  [#permalink]

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New post 01 Jul 2015, 10:21
are you considering only distinct factors?
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Re: M01-35  [#permalink]

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New post 01 Jul 2015, 11:49
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Re M01-35  [#permalink]

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New post 29 Jul 2017, 00:21
I think this is a high-quality question and I agree with explanation.
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M01-35  [#permalink]

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New post 31 Jul 2018, 08:45
Bunuel niks18 chetan2u KarishmaB GMATPrepNow pushpitkc generis

Quote:
If \(@x\) is the number of distinct positive divisors of \(x\), what is the value of \(@(@90)\)?


Quote:
Finding the Number of Factors of an Integer

First 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.


Correct, but is not question asking us the distinct number of factors and not a TOTAL number of factors?

90 can be factorized in terms of primes as \(3^2\) , 2 and 5
UNIQUE Factors will be only 2,3 and 5. Am I correct?
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Re: M01-35  [#permalink]

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New post 31 Jul 2018, 08:49
1
adkikani wrote:
Bunuel niks18 chetan2u KarishmaB GMATPrepNow pushpitkc generis

Quote:
If \(@x\) is the number of distinct positive divisors of \(x\), what is the value of \(@(@90)\)?


Quote:
Finding the Number of Factors of an Integer

First 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.


Correct, but is not question asking us the distinct number of factors and not a TOTAL number of factors?

90 can be factorized in terms of primes as \(3^2\) , 2 and 5
UNIQUE Factors will be only 2,3 and 5. Am I correct?


Those are prime factors.

Distinct number of factors is the same as the total number of factors.
_________________

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Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
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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.


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Re: M01-35  [#permalink]

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New post 01 Aug 2018, 03:57
1
adkikani wrote:
Bunuel niks18 chetan2u KarishmaB GMATPrepNow pushpitkc generis

Quote:
If \(@x\) is the number of distinct positive divisors of \(x\), what is the value of \(@(@90)\)?


Quote:
Finding the Number of Factors of an Integer

First 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.


Correct, but is not question asking us the distinct number of factors and not a TOTAL number of factors?

90 can be factorized in terms of primes as \(3^2\) , 2 and 5
UNIQUE Factors will be only 2,3 and 5. Am I correct?


How about 6? Isn't that a unique factor too? What about 15? etc
As Bunuel said, no of factors is the same as unique factors. Note that we do not count 3 twice

\(90 = 2*3^2 * 5\)

Distinct Prime factors are 2, 3 and 5.

All unique factors are
1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Total 12 factors
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Re: M01-35 &nbs [#permalink] 01 Aug 2018, 03:57
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