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Manager  Joined: 27 May 2008
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The function f(n) = the number of factors of n. If p and q  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 63% (02:02) correct 37% (01:59) wrong based on 452 sessions

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The function f(n) = the number of factors of n. If p and q are positive integers and f(pq) = 4, what is the value of p?

(1) p + q is an odd integer

(2) q is less than p

Originally posted by selvae on 05 Dec 2008, 20:30.
Last edited by Bunuel on 16 May 2014, 05:24, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: The function f(n) = the number of factors of n. If p and q  [#permalink]

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5
4
Hi All,

Certain DS questions will be easier to handle if you do a bit of 'up-front' work with what you're given in the prompt (before you include the extra information that's given in the two Facts).

Here, we're told that the function f(N) = the number of FACTORS of N. We're also told that P and Q are POSITIVE INTEGERS and f(PQ) = 4. We're asked for the value of P.

Let's ignore the P and Q for a moment and just focus on the one thing....finding an example of a number that has 4 factors....And let's go for a simple one...In this way, we're just TESTing VALUES...

6 --> 1, 2, 3 and 6

This example means that f(PQ) = f(6) = 4, so P and Q can be some combination of 1 and 6 OR 2 and 3

There are other numbers we can use besides 6, of course, but I'm just going to think about this one example as I work with the two Facts...

Fact 1: P + Q is an ODD integer

Using our prior work....
IF....P = 3 and Q = 2 ...the answer to the question is 3
IF....P = 6 and Q = 1 ...the answer to the question is 6
Fact 1 is INSUFFICIENT

Fact 2: Q is less than P

We can actually use the SAME two TESTs that we used in Fact 1....

IF....P = 3 and Q = 2 ...the answer to the question is 3
IF....P = 6 and Q = 1 ...the answer to the question is 6
Fact 2 is INSUFFICIENT

Combined, we already have two different solutions that 'fit' both Facts...
Combined, INSUFFICIENT

GMAT assassins aren't born, they're made,
Rich
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##### General Discussion
Intern  Joined: 05 Dec 2008
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1
f(pq) = 4 means that the product pq has 4 factors. What does this tell us about p and q?
To begin, we might recognize that if the product pq has 4 factors, then p and q must each have 4 factors or fewer.
Under what circumstances could one value, say p, have 4 factors? When q = 1 and p has 4 factors.
For example, p=6 and q=1 or p=21 and q=1. (or we could have p=1 and q=6 etc)
If one value is 1 then the other value must have 4 factors.

What about situations where neither p nor q is equal to one. In these cases, p and q will have 2 or more factors.
What sorts of numbers have exactly 2 factors? Prime numbers.
So, if p=3 and q=5 (both primes) then the product pq will have 4 factors (i.e., f(15)=4

We've deduced a fair bit here already about p and q, so let's check the data.

(1) p+q is an odd integer
Using the above deductions alone, we see that p could be 1, or p could be 3 or 5 or . . .etc.
Not enough information. We can't definitively determine the value of p.

(2) q is less than p
Not enough info (see above scenarios). We can't definitively determine the value of p.

Using (1) and (2) still leads us to the conclusion that there is not enough information.
We could have q=1 and p=21 or we could have q=3 and p=5

Manager  Joined: 30 Sep 2008
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f(pq) = 4 => pq has 4 factors: p, q, 1, pq => p, q are prime

(1) p + q is an odd integer - every prime is odd, except 2, so this doesn't give more info
(2) q is less than p <-- insuff

(1) + (2) <-- insuff, hence E
Intern  Joined: 05 Dec 2008
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The last post is not entirely correct.
"f(pq) = 4 => pq has 4 factors: p, q, 1, pq => p, q are prime"

We can't conclude that p and q are prime. What if p=1?
Then we could have p=1 and q=14.
It still holds that f(pq)=4, but in this case q is not prime and p is not prime either.

The answer is still E, but the poster would have gotten the answer incorrect if statement (1) had been "p and q are both primes"
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selvae wrote:
The function f(n) = the number of factors of n. If p and q are positive integers and f(pq) = 4, what is the value of p?

(1) p + q is an odd integer
(2) q is less than p

since f(pq) = 4, f(pq) could be 6 or 8 or 10 or so on.

1: if p is odd q is even and vice versa.
if f(pq) = 6, p could be 1 and q = 6 or p = 6 and q=1.
nsf...

2: q<p.
q = 1, p = 6.
q = 2, p = 4. not suff.

1 and 2 also nsf... E.
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Re: The function f(n) = the number of factors of n. If p and q  [#permalink]

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The question should be reworded to state pq is product of p & q...I took it for a 2 digit number pq..
Although I got the same answer Math Expert V
Joined: 02 Sep 2009
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Re: The function f(n) = the number of factors of n. If p and q  [#permalink]

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JusTLucK04 wrote:
The function f(n) = the number of factors of n. If p and q are positive integers and f(pq) = 4, what is the value of p?

(1) p + q is an odd integer

(2) q is less than p

The question should be reworded to state pq is product of p & q...I took it for a 2 digit number pq..
Although I got the same answer If it were a two-digit number pq, then it would be explicitly stated.
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Re: The function f(n) = the number of factors of n. If p and q  [#permalink]

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Bunuel wrote:
JusTLucK04 wrote:
The function f(n) = the number of factors of n. If p and q are positive integers and f(pq) = 4, what is the value of p?

(1) p + q is an odd integer

(2) q is less than p

The question should be reworded to state pq is product of p & q...I took it for a 2 digit number pq..
Although I got the same answer If it were a two-digit number pq, then it would be explicitly stated.

--------------------------------------------
hi Bunuel,
can you please explain the correct method to solve this problem?

Regards
Harish
Intern  Joined: 23 Apr 2014
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The function f(n) = the number of factors of n. If p and q  [#permalink]

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Following few things can be established from the question
f(pq)=4,meaning the number of factors of pq is 4 which can only happen in following cases
↪p^1 q^1=where p and q are two different primes
↪p^0 q^3=where p is 1 and q is any prime

Condition 1
p+q=an odd integer which can only happen in following cases
↪p(Even prime-2)+q(odd prime)=odd integer→does reflect on the value of p but p could take other values….not sufficient
↪p(1)+q(even prime)=odd integer→does reflect on the value of p but p could take other values…not sufficient
↪p(odd prime)+q(even prime-2)=odd integer→ p can take more than one values……not sufficient

Condition 2
↪p>q…..p can take up any value, it can be 1 or any prime…..not sufficient

Combining (1)+(2)since p>q it is evident that
↪① p can^' t be 1→therefore pq can' t take the form of p^0.q^3
it can only take form of p^1.q^1 (for two distinct primes p and q)

↪② it is established that p and q are both prime integers and sum of two prime integer is an odd integer
↪meaning either p or q is even prime 2
↪since p>q and sum of prime p and q is odd,then q=2
↪p is an odd prime and it can take up any value……not sufficient

Originally posted by sam26can on 25 Jun 2014, 22:48.
Last edited by sam26can on 26 Jun 2014, 00:11, edited 2 times in total.
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The function f(n) = the number of factors of n. If p and q  [#permalink]

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The number of factors of f(p^m*q^n) can be obtained by product of (m+1)(n+1).

As given in the problem statement f(pq) = 4 that means (m+1)(n+1) = 4. The only way to achieve is 2x2 with m, n > 0 -> m & n be 1,1.

So p & q are single exponent prime numbers.

From I:
p+q=an odd integer which can only happen in following cases:
i)p->Odd & q->Even
ii)p->Even & q->Odd

As p,q are prime number and exactly one of two must be even so either p or q but not both is 2 -> Not sufficient

From II:
q is less than p that means q can be 2 and p can be 3 OR q can be 3 and p can be 5 -> Not sufficient.

I + II:
q has to be 2 but no value can be given to p. it can take any odd prime number value. So answer should be (E)

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Re: The function f(n) = the number of factors of n. If p and q  [#permalink]

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1
one way to look the solution is :

F(p q) = 4 , that means total factors of P & q product is 4. When is this possible.

to calculate total factors including PQ & 1 we split the product in prime factors : 2^a * 3^b * 5^c ........
then total factors should be (a+1) *(b+1) * (C+1)....... ----> this product should be equal to 4----> means either a and b =1 or b&c=1 or any two powers of prime number are 1 and rest are zero.

Now look at combinations we can make to give f (p q) =4

1. 2^1 x 3^1 ; total factors 4
2. 3^1 x 5^1 ; total factors 4
3. 2^3 ; total factors 4
4. and so........ u can make infinite combinations.... therefore can't tell value of P : it can be 2,3,5,8,7,.....

We are given that P+Q is odd. there one of them P or Q has to be even. This only tells us that one of the prime number is 2 the other is unknown. therefore insufficient.

We are given that P>q, then p= 8 and q=1 or P=5 and q=3...... so on...therefore insufficient.

Combine 1+2
we have P+q= odd and P>Q

so we can have combination like

p=3 and q=2
p=8 and q =1
p=5 and q=2

therefore insufficient.

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Re: The function f(n) = the number of factors of n. If p and q  [#permalink]

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For statement 1, let us assume p=7, q=8 & p=12, q=9 [15 & 21 have four factors each]. Therefore, p cannot be determined from the information.

For statement 2, assume the same values, q=7 & q=9. Insufficient information.

As multiple values can be obtained despite taking both statements, correct answer is 'E'.
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The function f(n) = the number of factors of n. If p and q  [#permalink]

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Any product pq must have the following factors: {1, p, q, and pq}. If the product pq has no additional factors, the p and q must be prime.

Statement (1) tells us that the sum of p and q is an odd integer. Therefore either p or q must be even, while the other is odd. Since we know p and q are prime, either p or q must be equal to 2 (the only even prime number). However, statement (1) does not provide enough information for us to know which of the variables, p and q, is equal to 2.

Statement (2) tells us that q < p, which does not give us any information about the value of p.

From both statements taken together, we know that, since 2 is the smallest prime number, q must equal 2. However, we cannot determine the value of p.
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Top Contributor
selvae wrote:
The function f(n) = the number of factors of n. If p and q are positive integers and f(pq) = 4, what is the value of p?

(1) p + q is an odd integer

(2) q is less than p

Target question: What is the value of p?

Given: f(n) = the number of factors of n. If p and q are positive integers and f(pq) = 4

Statement 1: p + q is an odd integer
Here are two sets of values for p and q that satisfy statement 1:
Case a: p = 2 and q = 3. Notice that the product, 6, has 4 factors: 1, 2, 3 and 6. In this case, p = 2
Case b: p = 7 and q = 2. Notice that the product, 14, has 4 factors: 1, 2, 7 and 14. In this case, p = 7
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: q is less than p
From the given information, we know that p and q are both prime numbers.
Here are two sets of values for p and q that satisfy statement 2:
Case a: p = 3 and q = 2. Notice that the product, 6, has 4 factors: 1, 2, 3 and 6. In this case, p = 3
Case b: p = 7 and q = 2. Notice that the product, 14, has 4 factors: 1, 2, 7 and 14. In this case, p = 7
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
There are several values of p and q that satisfy BOTH statements. Here are two:
Case a: p = 3 and q = 2. Notice that the product, 6, has 4 factors: 1, 2, 3 and 6. In this case, p = 3
Case b: p = 7 and q = 2. Notice that the product, 14, has 4 factors: 1, 2, 7 and 14. In this case, p = 7
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Cheers,
Brent
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Re: The function f(n) = the number of factors of n. If p and q  [#permalink]

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