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If p and q are prime numbers, is pq+1 an odd number?

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If p and q are prime numbers, is pq+1 an odd number?  [#permalink]

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New post 04 Oct 2018, 00:55
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[Math Revolution GMAT math practice question]

If \(p\) and \(q\) are prime numbers, is \(pq+1\) an odd number?

\(1) p – q = 5\)
\(2) p = 7\)

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If p and q are prime numbers, is pq+1 an odd number?  [#permalink]

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New post 04 Oct 2018, 01:46
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Statment 1 Since p and q is prime, and 2 is the only even prime number, difference between 2 prime number is only possible if 1 of the number is 2 (even). Sufficient to answer whether odd*even+1= odd or not.

Statemnt 2= P=7.... q can be even or odd, not enough.
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Re: If p and q are prime numbers, is pq+1 an odd number?  [#permalink]

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New post 04 Oct 2018, 06:27
MathRevolution wrote:
[Math Revolution GMAT math practice question]

If \(p\) and \(q\) are prime numbers, is \(pq+1\) an odd number?

\(1) p – q = 5\)
\(2) p = 7\)

\(p,q\,\,{\rm{primes}}\,\,\,\,\left( * \right)\)

\(pq + 1\,\,\mathop = \limits^? \,\,{\text{odd}}\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\boxed{\,?\,\,\,:\,\,\,\,p = 2\,\,\,{\text{or}}\,\,\,q = 2\,\,\,}\)

\(\left( 1 \right)\,\,p - q = 5\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle\)

\(\left( {**} \right)\,\,p \ne 2\,\,{\rm{and}}\,\,q \ne 2\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,p,q\,\,\,{\rm{odd}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {5 = } \right)\,\,p - q\,\,{\rm{even}}\,\,,\,\,\,{\rm{impossible}}\)


\(\left( 2 \right)\,\,p = 7\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {p,q} \right) = \left( {7,2} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {p,q} \right) = \left( {7,3} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\,\,\)



This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: If p and q are prime numbers, is pq+1 an odd number?  [#permalink]

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New post 04 Oct 2018, 08:55
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MathRevolution wrote:
[Math Revolution GMAT math practice question]

If \(p\) and \(q\) are prime numbers, is \(pq+1\) an odd number?

\(1) p – q = 5\)
\(2) p = 7\)


Given: p and q are prime numbers

Target question: Is pq + 1 an odd number?
This is a good candidate for rephrasing the target question.
In order for pq + 1 to be odd, we need pq to be EVEN
So, we COULD rephrase our target question as Is pq an even number?, but we can do even better than that.
If p and q are PRIME numbers, and pq is even, then one of the values (p or q) must be 2. So, lets' rephrase our target question as follows:
REPHRASED target question: Is either p or q equal 2?

Aside: the video below has tips on rephrasing the target question

Statement 1: p – q = 5
The only way that p – q = 5 is if p = 7 and q = 2.
How do I know this?
Well, as you know, most prime numbers are odd. If p and q were both odd, we'd have: ODD - ODD = 5, which is impossible (ODD - ODD always equals EVEN)
So, it CANNOT be the case that p and q are both odd.
In other words, it must be the case that p or q is EVEN.
So, the answer to the REPHRASED target question is YES, either p or q DOES equal 2
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: p = 7
Since we don't have any information, we can't answer the REPHRASED target question with certainty.
So, statement 2 is SUFFICIENT
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

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Re: If p and q are prime numbers, is pq+1 an odd number?  [#permalink]

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New post 07 Oct 2018, 18:33
=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Modifying the question:
\(pq + 1\) is odd only when \(pq\) is even. So, the question is asking whether either \(p\) or \(q\) is an even prime number. Since the only even prime number is \(2\), the question is asking whether \(p\) or \(q\) is equal to \(2\).


Condition 1):
For \(p – q = 5\), either \(p\) or \(q\) must be even. Since the only even prime number is \(2\), we must have \(p = 7\) and \(q = 2\).
Thus, condition 1) is sufficient.

Condition 2)
Since it provides no information about \(q\), condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A
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Re: If p and q are prime numbers, is pq+1 an odd number? &nbs [#permalink] 07 Oct 2018, 18:33
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