If p and q are positive integers such that pq has exactly 4 unique pos

Question

TRICKY QUESTION

If p and q are positive integers such that pq has exactly 4 unique positive factors and  p < q  , what is the value of the integer p?
 
(1) The sum of p and q is an odd integer.
(2) The difference between p and q is 1.

Source : Manhattan Advanced Quant Practice Problems

attari92 · Answer

The question stem boils down to that integers 'p' and 'q' are prime and we need to figure out the value of 'p' given p<q.
 statement 1. is explicitly not sufficient as (2+3),(2+5)....
 
 statement 2. implies p=2 and q=3, so sufficient

sureshhbs · Answer

But the question says it is p<Q. So p will be always 2

Bunuel · Answer

If p and q are positive integers such that pq has exactly 4 unique positive factors and  p < q  , what is the value of the integer p?

Given that pq has exactly 4. We can have the following two cases:

(i)   p = 1  and  q = (prime)^3  -->  pq = 1*(prime)^3  --> the number of factors = 3 + 1 = 4. For example,  pq = 1*2^3 = 8 .

(ii)   p = prime_1  and  q = prime_2  -->  pq = prime_1*prime_2  --> the number of factors = (1 + 1)(1 + 1) = 4. For example,  pq = 2*3 = 6 .

As we can see p is either 1 or some prime.

(1) The sum of p and q is an odd integer. For the sum of two integers to be odd one of them must be odd and another must be even. We cannot get the unique value of p. For example:
p = 1 and q = 2^3 = 8
p = 2 and q = 3.

Not sufficient.

(2) The difference between p and q is 1. Can we have case  (i)  here? No. If p = 1 and  q = (prime)^3 , the the least difference between p and q is 2^3 - 1 = 7. Therefore both p and q must be primes. The only primes with difference of 1 are 2 and 3. Sufficient.

Answer: B.

Hope it's clear.

Lepe96 · Answer

Hi actually It isn't clear why wouldn't it be sufficient that the 2 numbers are primes? I mean any product between 2 primes has 4 unique factors

In the e.g. given p,q primes so "pq" would have 4 factors namely 1,p,q,pq

So once we know that the sum is odd we can deduce that 1 of the 2 is even, and being p,q primes and knowing that p < q we can deduce that p=2

Please show me the flaw in my reasoning...

Bunuel · Answer

Please read the post just above yours:

Given that pq has exactly 4. We can have the following  two  cases:

(i)   p = 1  and  q = (prime)^3  -->  pq = 1*(prime)^3  --> the number of factors = 3 + 1 = 4. For example,  pq = 1*2^3 = 8 . 
 (ii)   p = prime_1  and  q = prime_2  -->  pq = prime_1*prime_2  --> the number of factors = (1 + 1)(1 + 1) = 4. For example,  pq = 2*3 = 6 .

As we can see  p is either 1  or some prime.

(1) The sum of p and q is an odd integer. For the sum of two integers to be odd one of them must be odd and another must be even. We cannot get the unique value of p. For example:
 p = 1 and q = 2^3 = 8 
p = 2 and q = 3.

Not sufficient.

If p and q are positive integers such that pq has exactly 4 unique pos

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia