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

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Wed Oct 03, 2018 11:48 pm

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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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Source: Beat The GMAT — Data Sufficiency | Wed Oct 03, 2018 11:48 pm

fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Thu Oct 04, 2018 5:19 am

Max@Math Revolution 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

$$p,q\,\,{\rm{primes}}\,\,\,\,\left( * \right)$$
$$pq + 1\,\,\mathop = \limits^? \,\,{\text{odd}}\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\boxed{\,?\,\,\,:\,\,\,\,p = 2\,\,\,{\text{or}}\,\,\,q = 2\,\,\,}$$
$$\left( 1 \right)\,\,p - q = 5\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
$$\left( {**} \right)\,\,p \ne 2\,\,{\text{and}}\,\,q \ne 2\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,p,q\,\,\,{\text{odd}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {5 = } \right)\,\,p - q\,\,{\text{even}}\,\,,\,\,\,{\text{impossible}}$$
$$\left( 2 \right)\,\,p = 7\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {p,q} \right) = \left( {7,2} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {p,q} \right) = \left( {7,3} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\,\,$$

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

Regards,
Fabio.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Oct 04, 2018 7:43 am

Max@Math Revolution wrote: 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: Here's a video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

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

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pmFollowed by: 5 members

by deloitte247 » Thu Oct 04, 2018 1:26 pm

Question = Is pq + 1 = odd number?

Statement 1 = p - q = 5
p = 5 + q
From question, pq + 1 = odd number.
(5 + q ) * q + 1 = ?
$$5q\ +\ q^2\ +\ 1=?$$
There is no specific information about the value of q, hence statement 1 is INSUFFICIENT.

Statement 2 = p = 7
From question pq + 1 = odd number
7q + 1 = ?
There is no specific information about the value of q, hence statement 2 is INSUFFICIENT.
COMBINING STATEMENT 1 AND 2 TOGETHER.
$$\left(5q\ +\ q^2\ +\ 1\right)+\left(7q\ +\ 1\right)=0$$
$$12q\ +\ q^2\ +\ 2=0$$
$$\ q^2\ +\ 12q\ +\ 2=0$$
Using the quadratic information, $$x\ =\frac{\left(-b\ \frac{+}{-}\sqrt{b^2-4ac}\right)}{2a}$$
Where a = 1, b = 12, c = 2 and x = q
$$q\ =\frac{\left(-12\ \frac{+}{-}\sqrt{12^2-\left(4\ \cdot\ 1\ \cdot\ 2\right)}\right)}{2\ \cdot\ 1}$$
$$q\ =\frac{\left(-12\ \frac{+}{-}\sqrt{144\ -\ 8}\right)}{2\ }$$
$$q\ =\frac{\left(-12\ \frac{+}{-}\sqrt{144\ -\sqrt{8}\ }\right)}{2\ }$$
$$q\ =\frac{\left(-12\ \frac{+}{-}12-\sqrt{8}\ \right)}{2\ }$$
$$q\ =\frac{\left(-12\ \frac{+}{-}12-\sqrt{8}\ \right)}{2\ }\ or\ q\ =\ \frac{\left(-12\ -12\ -\sqrt{8}\right)}{2}$$
$$q\ =\frac{-\sqrt{8}\ }{2\ }\ or\ q\ =\ \frac{\left(-24\ -\sqrt{8}\right)}{2}$$
$$q\ =\frac{-\sqrt{8}\ }{2\ }\ or\ q\ =\ -12\ -\sqrt{8}$$
After combining statement 1 and 2 together, the value of q is not yet specific, hence the two statement together are not SUFFICIENT

Option E is CORRECT.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Oct 04, 2018 1:35 pm

Be careful, deloitte247
You haven't used the information about p and q being prime numbers.

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Sun Oct 07, 2018 5:20 pm

=>

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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