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BillyZ
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Joined: 14 Nov 2016
Last visit: 24 Jan 2026
Posts: 1,135
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Location: Malaysia
Concentration: General Management, Strategy
Schools: Sloan '23 (D) Tuck '23 (D) Darden '23 (D) Ross '23 (D) Kellogg '23 (D) Wharton '23 (D) Booth '23 (D) Fuqua '24 (A) Harvard '24 (D) Stanford '24 (D)
GMAT 1: 750 Q51 V40 (Online)
GPA: 3.53
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Schools: Sloan '23 (D) Tuck '23 (D) Darden '23 (D) Ross '23 (D) Kellogg '23 (D) Wharton '23 (D) Booth '23 (D) Fuqua '24 (A) Harvard '24 (D) Stanford '24 (D)
GMAT 1: 750 Q51 V40 (Online)
Posts: 1,135
B
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Difficulty:
95%
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Question Stats:
42% (02:07) correct
58%
(02:06)
wrong
based on 212
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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
27 Mar 2017, 01:01
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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.
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.
General Discussion
27 Mar 2017, 00:07
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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
statement 1. is explicitly not sufficient as (2+3),(2+5)....
statement 2. implies p=2 and q=3, so sufficient
