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

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If p and q are positive integers such that pq has exactly 4 unique pos  [#permalink]

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Updated on: 27 Mar 2017, 01:35
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95% (hard)

Question Stats:

39% (02:06) correct 61% (02:00) wrong based on 100 sessions

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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

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Originally posted by hazelnut on 26 Mar 2017, 23:49.
Last edited by hazelnut on 27 Mar 2017, 01:35, edited 1 time in total.
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Re: If p and q are positive integers such that pq has exactly 4 unique pos  [#permalink]

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27 Mar 2017, 01:01
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1
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.
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Re: If p and q are positive integers such that pq has exactly 4 unique pos  [#permalink]

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27 Mar 2017, 00:07
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
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Re: If p and q are positive integers such that pq has exactly 4 unique pos  [#permalink]

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27 Mar 2017, 00:20
attari92 wrote:
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

But the question says it is p<Q. So p will be always 2
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Re: If p and q are positive integers such that pq has exactly 4 unique pos  [#permalink]

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27 Sep 2017, 01:02
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...
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Posts: 49968
Re: If p and q are positive integers such that pq has exactly 4 unique pos  [#permalink]

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27 Sep 2017, 01:07
Lepe96 wrote:
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...

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.
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Re: If p and q are positive integers such that pq has exactly 4 unique pos &nbs [#permalink] 27 Sep 2017, 01:07
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# If p and q are positive integers such that pq has exactly 4 unique pos

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