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# M01-35

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Math Expert
Joined: 02 Sep 2009
Posts: 49968

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16 Sep 2014, 00:16
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Difficulty:

35% (medium)

Question Stats:

69% (00:46) correct 31% (01:00) wrong based on 176 sessions

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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

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Math Expert
Joined: 02 Sep 2009
Posts: 49968

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16 Sep 2014, 00:16
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: $$(2+1)(1+1)=6$$.

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Joined: 15 Apr 2015
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01 Jul 2015, 11:21
are you considering only distinct factors?
Math Expert
Joined: 02 Sep 2009
Posts: 49968

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01 Jul 2015, 12:49
SoumiyaGoutham wrote:
are you considering only distinct factors?

Well, yes. How else?
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Joined: 09 Jul 2017
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29 Jul 2017, 01:21
I think this is a high-quality question and I agree with explanation.
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Joined: 04 Sep 2016
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Location: India
WE: Engineering (Other)

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31 Jul 2018, 09:45
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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Math Expert
Joined: 02 Sep 2009
Posts: 49968

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31 Jul 2018, 09:49
1
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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Joined: 16 Oct 2010
Posts: 8386
Location: Pune, India

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01 Aug 2018, 04:57
1
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, 04:57
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# M01-35

Moderators: chetan2u, Bunuel

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