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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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New post 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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New post 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.

Answer: E.

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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New post 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.


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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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New post 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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New post 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.

Answer: E.

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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New post 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.

Answer: E.

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

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New post 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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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

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