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

Question

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

Bunuel · Accepted Answer

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.

Lucky2783 · Answer

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
Consider +1 Kudos if you do not hate this post.

TARGET730 · Answer

Thanks bunuel for the solution and the alternate question.

chetan2u · Answer

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..

Bunuel · Answer

Yes, that's true. Two cases were enough to get E as the answer, so did not have to look for the third case.

Nunuboy1994 · Answer

"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?

Bunuel · Answer

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.

KrishnakumarKA1 · Answer

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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A function is defined as f(n) = the number of factors of n. If f(pqr

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