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# A function is defined as f(n) = the number of factors of n. If f(p*q*r

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A function is defined as f(n) = the number of factors of n. If f(p*q*r [#permalink]

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12 Mar 2015, 03:02
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A function is defined as f(n) = the number of factors of n. If f(p*q*r) = 8, where p, q and r are positive integers, what is the value of p?

(1) p + q + r is an even number
(2) q < p < r

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A function is defined as f(n) = the number of factors of n. If f(p*q*r [#permalink]

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12 Mar 2015, 03:50
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TARGET730 wrote:
A function is defined as f(n) = the number of factors of n. If f(p*q*r) = 8, where p, q and r are positive integers, what is the value of p?

(1) p + q + r is an even number
(2) q < p < r

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.

A function is defined as f(n) = the number of factors of n. If f(p*q*r) = 8, where p, q and r are positive integers, what is the value of p?

According to the theory bit above, for an integer (p*q*r) to have 8 factors it should be of a form of prime^7 (number of factors 1+7=8) or prime*prime^3 (number of factors (1+1)(1+3)=8) or prime*prime*prime (number of factors (1+1)(1+1)(1+1)=8).

(1) p + q + r is an even number.

(2) q < p < r

Even when we take the statements together we cannot get the value of p. For example,
q = 1, p = 2, r = 3^3 --> pqr = 2*3^3 --> number of factors = 8.
q = 2, p = 3, r = 3^2 --> pqr = 2*3^3 --> number of factors = 8.

Similar question to practice: the-function-f-n-the-number-of-factors-of-n-if-p-and-q-73680.html

Hope it helps.
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A function is defined as f(n) = the number of factors of n. If f(p*q*r [#permalink]

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Updated on: 15 Mar 2015, 07:30
TARGET730 wrote:
A function is defined as f(n) = the number of factors of n. If f(p*q*r) = 8, where p, q and r are positive integers, what is the value of p?

(1) p + q + r is an even number
(2) q < p < r

Option A:
q=2
p=3
r=7
F(2.3.7) = 8 (hope everyone knows how to calculate total factors of a number, if not the bunuel's math book is very handy , please refer that )

another such set of values is :
q=2
p=3
r=5
F(2.3.5) = 8

option A : NS

Option B:
q<p<r :- in both examples above q<p<r so this option is also NS.

Together as well , it makes no difference . Option E is right choice.

Thanks
Lucky
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Originally posted by Lucky2783 on 15 Mar 2015, 05:50.
Last edited by Lucky2783 on 15 Mar 2015, 07:30, edited 1 time in total.
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Re: A function is defined as f(n) = the number of factors of n. If f(p*q*r [#permalink]

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15 Mar 2015, 06:00
Thanks bunuel for the solution and the alternate question.
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Re: A function is defined as f(n) = the number of factors of n. If f(p*q*r [#permalink]

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15 Mar 2015, 06:38
1
1
Bunuel wrote:
TARGET730 wrote:
A function is defined as f(n) = the number of factors of n. If f(p*q*r) = 8, where p, q and r are positive integers, what is the value of p?

(1) p + q + r is an even number
(2) q < p < r

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.

A function is defined as f(n) = the number of factors of n. If f(p*q*r) = 8, where p, q and r are positive integers, what is the value of p?

According to the theory bit above, for an integer (p*q*r) to have 8 factors it should be of a form of either prime^7 (number of factors 1+7=8) or prime*prime^3 (number of factors (1+1)(1+3)=8).

(1) p + q + r is an even number.

(2) q < p < r

Even when we take the statements together we cannot get the value of p. For example,
q = 1, p = 2, r = 3^3 --> pqr = 2*3^3 --> number of factors = 8.
q = 2, p = 3, r = 3^2 --> pqr = 2*3^3 --> number of factors = 8.

Similar question to practice: the-function-f-n-the-number-of-factors-of-n-if-p-and-q-73680.html

Hope it helps.

hi bunuel,
although, it does not change the answer but there is one more way of getting 8 factors.. prime*prime*prime..
no of factors=(1+1)(1+1)(1+1)=8..
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Re: A function is defined as f(n) = the number of factors of n. If f(p*q*r [#permalink]

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15 Mar 2015, 06:54
chetan2u wrote:
Bunuel wrote:
TARGET730 wrote:
A function is defined as f(n) = the number of factors of n. If f(p*q*r) = 8, where p, q and r are positive integers, what is the value of p?

(1) p + q + r is an even number
(2) q < p < r

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.

A function is defined as f(n) = the number of factors of n. If f(p*q*r) = 8, where p, q and r are positive integers, what is the value of p?

According to the theory bit above, for an integer (p*q*r) to have 8 factors it should be of a form of either prime^7 (number of factors 1+7=8) or prime*prime^3 (number of factors (1+1)(1+3)=8).

(1) p + q + r is an even number.

(2) q < p < r

Even when we take the statements together we cannot get the value of p. For example,
q = 1, p = 2, r = 3^3 --> pqr = 2*3^3 --> number of factors = 8.
q = 2, p = 3, r = 3^2 --> pqr = 2*3^3 --> number of factors = 8.

Similar question to practice: the-function-f-n-the-number-of-factors-of-n-if-p-and-q-73680.html

Hope it helps.

hi bunuel,
although, it does not change the answer but there is one more way of getting 8 factors.. prime*prime*prime..
no of factors=(1+1)(1+1)(1+1)=8..

Yes, that's true. Two cases were enough to get E as the answer, so did not have to look for the third case.
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Re: A function is defined as f(n) = the number of factors of n. If f(p*q*r [#permalink]

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22 Feb 2017, 16:43
"According to the theory bit above, for an integer (p*q*r) to have 8 factors it should be of a form of prime^7 (number of factors 1+7=8) or prime*prime^3 (number of factors (1+1)(1+3)=8) or prime*prime*prime (number of factors (1+1)(1+1)(1+1)=8)."

What do you mean by this? I don't understand what you mean for an integer (p x q x r) to have 8 factors it should be a form of prime^7. What is prime^7?
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Joined: 02 Sep 2009
Posts: 46335
Re: A function is defined as f(n) = the number of factors of n. If f(p*q*r [#permalink]

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23 Feb 2017, 00:37
Nunuboy1994 wrote:
"According to the theory bit above, for an integer (p*q*r) to have 8 factors it should be of a form of prime^7 (number of factors 1+7=8) or prime*prime^3 (number of factors (1+1)(1+3)=8) or prime*prime*prime (number of factors (1+1)(1+1)(1+1)=8)."

What do you mean by this? I don't understand what you mean for an integer (p x q x r) to have 8 factors it should be a form of prime^7. What is prime^7?

prime^7 as a prime number in 7th power. For example, 3^7. Any prime in 7th power will have 8 factors, which is explained in previous posts:

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.
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Re: A function is defined as f(n) = the number of factors of n. If f(p*q*r [#permalink]

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23 Feb 2017, 03:31
Prompt analysis

f(n) = number of factors of n and f(p*q*r) = 8. that means p*q*r will be in form of a*b*c or d^3 * e or f^7, where a,b,c,d,e,f are prime numbers.

Superset
the value of p will be a positive integer

Translation
In order to find the value of p, we need:
1# exact value of p, q, r.
2# three equations in these three variables.
3# any other property of condition to restrict the value of p,q and r.

Statement analysis

St 1: p +q+ r is even. It can be even if the sum is in the form of odd+odd+even or even + even + even. possible conditions could be 2,3,5 or 2,2,6 or 2,4,16. we can only infer that one of the number will be 2. INSUFFICIENT

St 2: again if we take above mentioned cases i.e. 2,3,5 or 2,2,6 or 2,4,16 some of them might violate the condition. INSUFFICIENT.

Option E
Re: A function is defined as f(n) = the number of factors of n. If f(p*q*r   [#permalink] 23 Feb 2017, 03:31
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