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The function f(n) for an integer n is defined as the number of  [#permalink]

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Question Stats: 64% (01:43) correct 36% (02:07) wrong based on 118 sessions

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The function f(n) for an integer n is defined as the number of positive integers that are factors of n.
For example, f(6) = 4 because 6 has 4 factors: 1, 2, 3, 6. What is the value of f(f(24^2))?

A. 2
B. 4
C. 6
D. 8
E. 12

Source: Experts Global

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Senior PS Moderator V
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The function f(n) for an integer n is defined as the number of  [#permalink]

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nigina93 wrote:
I would go with B and here is why.
First take care of f(24^2) or 576 which has 21 factors (before 24 it has 10 factors(1,2,3,4,6,8,9,12,16,18) + 1 (24) itself +10 factors after 24).
Now, we are left with f(21), which has 4 factors (1,3,7,21), asnwer is B.
I wonder whether there is easier way of realizing 21 factors of 576, because I do list them manually and took longer than 2 mins, Bunuel?

For any number, in order to find the total number of factors of any numbers, we need to do the following steps

1. Prime factorize number to the form $$a^x*b^y*c^z$$ where a,b,c are the prime numbers.
2. Total number of prime factors are (x+1)(y+1)(z+1)

Let's try and use this to find the number of factors of 576 which is $$24^2$$

Prime-factorizing $$(576)^2$$, we get $$(24^2)^2 = (2^3*3)^2 = 2^6*3^2$$.
The number of factors of $$(576)^2$$ can be given as $$(6+1)(2+1) = 7*3 = 21$$

Therefore, the value for $$f(f(24^2)) = f(21)$$ is 4(Option B)
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The function f(n) for an integer n is defined as the number of  [#permalink]

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B

24*24 has 21 factors
21 has 4 factors
Manager  P
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Re: The function f(n) for an integer n is defined as the number of  [#permalink]

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I would go with B and here is why.
First take care of f(24^2) or 576 which has 21 factors (before 24 it has 10 factors(1,2,3,4,6,8,9,12,16,18) + 1 (24) itself +10 factors after 24).
Now, we are left with f(21), which has 4 factors (1,3,7,21), asnwer is B.
I wonder whether there is easier way of realizing 21 factors of 576, because I do list them manually and took longer than 2 mins, Bunuel?
Intern  S
Joined: 23 Nov 2016
Posts: 10
The function f(n) for an integer n is defined as the number of  [#permalink]

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nigina93 wrote:
I would go with B and here is why.
First take care of f(24^2) or 576 which has 21 factors (before 24 it has 10 factors(1,2,3,4,6,8,9,12,16,18) + 1 (24) itself +10 factors after 24).
Now, we are left with f(21), which has 4 factors (1,3,7,21), asnwer is B.
I wonder whether there is easier way of realizing 21 factors of 576, because I do list them manually and took longer than 2 mins, Bunuel?

There IS an easier way.

Given $$f(f(24^2))$$
For simplicity's sake let $$f(24^2)$$=z
∴ we have f(z). Now lets solve z = $$f(24^2)$$ --> $$f( (8*3)^2 )$$ --> $$f((2^3*3)^2)$$ -- > $$f(2^6*3^2)$$
To get the number of factors just add 1 to the powers and multiply (6+1) (2+1) = 7*3
∴ z=7*3
f(z)= (1+1)(1+1) = 2*2 = 4
∵ $$(7^1)(3^1)$$

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Re: The function f(n) for an integer n is defined as the number of  [#permalink]

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Hey, guys. Thanks a lot, this way is much easier
Intern  B
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Re: The function f(n) for an integer n is defined as the number of  [#permalink]

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this question seems to be tough question
Intern  Joined: 15 May 2019
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Re: The function f(n) for an integer n is defined as the number of  [#permalink]

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nigina93 wrote:
Hey, guys. Thanks a lot, this way is much easier

How the answer can be B? yesterday it was C.
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Re: The function f(n) for an integer n is defined as the number of  [#permalink]

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Manually finding factors are pretty time concuming method.
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gmatjindi wrote:
Manually finding factors are pretty time concuming method.

gmatjindi - Unfortunately, thats the only way to solve this problem.

The important takeaway in this problem is

1. Prime factorize number to the form $$a^x∗b^y∗c^z$$ where a,b,c are the prime numbers.
2. Total number of prime factors can be calculated by using $$(x+1)(y+1)(z+1)$$
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Re: The function f(n) for an integer n is defined as the number of  [#permalink]

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Thank you very much for your detailed explanation! Re: The function f(n) for an integer n is defined as the number of   [#permalink] 20 May 2019, 14:13
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