Is pq = 1? (1) pqp = p (2) qpq = q

Question

Is pq = 1?
 
(1) pqp = p
 
(2) qpq = q

Bunuel · Accepted Answer

Notice that p=0 from the first statement does not exclude q being 0 too and similarly q=0 from the second statement does not exclude p being 0 too.

Is pq = 1?

(1) pqp = p --> p(pq-1)=0 -->  either  p=0 and q=(any number), including 0 so in this case pq=0 not 1  or  pq=1. Not sufficient.

(2) qpq = q --> q(pq-1)=0 -->  either  q=0 and p=(any number), including 0 so in this case pq=0 not 1  or  pq=1. Not sufficient.

(1)+(2) When combined we have that  either  p=q=0  or  pq=1. Not sufficient.

Answer: E.

When we consider two statements together we should take the values which satisfy both statements. For this question  pq=1  satisfies both statement, but  p=q=0  also satisfies both statements. So what you call "common solution" for this question is:  pq=1  OR  pq=0
eq{1} .

Hope it's clear.

fameatop · Answer

Statement 1- 
pqp = p
p (pq-1)=0 
It means p=0 or pq=1
Not sufficient

Statement 2- 
qpq = q
q (pq-1)=0 
It means q=0 or pq=1
Not sufficient

Statement 1 & 2- 
Combining both statements we get  p^3q^2  - p = 0
p( p^2q^2  - 1) = 0
p (pq-1)(pq+1) = 0
p=0 or pq=1 or pq=-1
Not sufficient
Answer E

Greenberg · Answer

Statement 1

pqp = p 
Dividing both sides in p
pq = 1

sufficient

Statement 2

qpq = q
Dividing both sides in q
qp = 1

sufficient

So the answer is (D)

IanStewart · Answer

It's certainly possible, using either or both statements, that p = q = pq = 1. It's also possible that p = q = pq = 0. So the answer is E.

DFG5150 · Answer

Since we don't know whether p=0 or q=0, we can't divide the ecuations by these variables.

Hades · Answer

You forgot to consider whether p or q are 0, in which case you'd be dividing by 0 (which is undefined). If it's not stated that the variable can't be 0, move the variables to one side and factor.

abilash10 · Answer

yes but if (1) gives us PQ= 1 as  a possible conclusion , and  so does  (2), then cant we conclude that PQ=1 ?

fameatop · Answer

That's not correct always and for the same reason i combined both the statements.

warabull · Answer

Hey can you please explain when to combine the statements and when not to ?
I was under the impression that for such questions, one needs to look at the common solution from both the qquestions to arrive at the answer.
If you could explain when to use these two approaches, it would help.

Thanks !

suk1234 · Answer

This is how I approached the question; if anyone can kindly point out my flaw.
The question is asking pq=1

or  p=1/q  or  q=1/p

Meaning  are p and q reciprocal? . Yes/No

Statement 1 
pqp=p
p^2q=p
q=p/p^2
q=1/p so Sufficient

Statement 2

qpq=q
q^2p=q
p=q/q^2
p=1/q so Sufficient

Therefore the answer is  C

Bunuel · Answer

If you say that each statement is sufficient, then you mean that the answer should be D, not C, right?

Now, the problem in your solution is that you cannot reduce pqp=p by p to get pq=1. This is because p can be zero and division be zero is not allowed.

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero.

So when you divide by p you assume, with no ground for it, that p does not equal to zero thus exclude a possible solution. Notice that p=0 and pq=1 both satisfy pqp=p.

The same logic applies to the second statement.

Hope it helps.

suk1234 · Answer

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero. Yeaps! got it now. Thanks for the reply, it's very helpful. And thanks for that important bit of information. :thumbup:

Qoofi · Answer

This is a very important rule. Thanks bunnel.

fskilnik · Answer

pq\,\,\mathop = \limits^? \,\,1

Let´s go straight to bifurcate (1+2), to guarantee the right answer is (E), indeed.

\left( {1 + 2} \right)\,\,\,\,\left\{ \matrix{
 \,{\rm{Take}}\,\,\left( {p,q} \right) = \left( {1,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr 
 \,{\rm{Take}}\,\,\left( {p,q} \right) = \left( {0,0} \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.

Bixy34 · Answer

Unable to understand how after combining p=0, q=0 is a valid case? Are we not supposed to take just the common solution which in this case is pq=1?

Bunuel · Answer

p=0 and q=0 satisfy both statements: p=0 makes (1) true, q=0 makes (2) true, and together they still satisfy both. In that case, pq=0, not 1. So pq is not uniquely determined, and the answer is E. Please review the solution here: https://gmatclub.com/forum/is-pq-1-1-pq ... l#p1272924 Also, pure algebraic questions are no longer a part of the DS syllabus of the GMAT. DS questions in GMAT Focus encompass various types of word problems, such as: Word Problems Work Problems Distance Problems Mixture Problems Percent and Interest Problems Overlapping Sets Problems Statistics Problems Combination and Probability Problems While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..." Check GMAT Syllabus for Focus Edition You can also visit the Data Sufficiency forum and filter questions by OG 2024-2025, GMAT Prep (Focus), and Data Insights Review 2024-2025 sources to see the types of questions currently tested on the GMAT. So, you can ignore this question. Hope it helps.

Is pq = 1? (1) pqp = p (2) qpq = q

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