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# What does p equal

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Manager
Joined: 15 Dec 2016
Posts: 102

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Updated on: 31 Oct 2017, 17:58
00:00

Difficulty:

85% (hard)

Question Stats:

23% (01:31) correct 77% (01:19) wrong based on 27 sessions

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What does p equal

(1) $$(\frac{p}{2}) ^ {(\frac{p}{2})} = \frac{p}{2}$$

(2) $$(\frac{2}{p}) ^ {(\frac{2}{p})} = \frac{2}{p}$$

I chose D but OA is E ...

Could you confirm how would you solve this ?

thank you !

Originally posted by jabhatta@umail.iu.edu on 02 Jul 2017, 18:38.
Last edited by broall on 31 Oct 2017, 17:58, edited 4 times in total.
Reformatted question
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Re: What does p equal  [#permalink]

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02 Jul 2017, 19:22
jabhatta@umail.iu.edu wrote:
What does p equal

(1) (P/2) ^ (P/2) = (P/2)

(2) (2/P) ^ (2/P) = (2/P)

I chose D but OA is E ...

Could you confirm how would you solve this ?

thank you !

Choose case if P= 2 or P= -2.....
Manager
Joined: 15 Dec 2016
Posts: 102
Re: What does p equal  [#permalink]

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02 Jul 2017, 20:05
Thanks Rohit ....

Question :

-- how long did it take you to solve this ?
-- did you try the strategy of just testing cases or did you think of some number properties theory or perhaps some theory regarding exponents

Have trouble determining if i should test cases in general or should i think of some theory to solve such problems

Thank you !
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02 Jul 2017, 20:39
jabhatta@umail.iu.edu wrote:
What does $$p$$ equal

(1) $$(\frac{p}{2}) ^ {(\frac{p}{2})} = \frac{p}{2}$$

(2) $$(\frac{2}{p}) ^ {(\frac{2}{p})} = \frac{2}{p}$$

I chose D but OA is E ...

Could you confirm how would you solve this ?

thank you !

To solve this question by algebra method:

Statement (1) :
To simplify the equation, set $$x=\frac{p}{2}$$. We have $$x^x=x$$

Note that $$x \neq 0$$ since we dont have $$0^0$$

If $$x>0$$, we have
$$x\ln x=\ln x \implies (x-1)\ln x=0 \implies x=1$$

If $$x<0$$, set $$t=-x \implies t>0$$, we have
$$(-t)^{(-t)}=-t \implies \frac{1}{(-t)^t}=-t \implies (-1)^{t+1} \times t^{t+1}=1 \implies t^{t+1}=(-1)^{t+1}$$
$$\implies t=-1 \implies x=-1$$

Hence $$x=\pm 1 \implies p=\pm 2$$

Insufficient.

Statement (2):

Solve the equation the same way as in (1), we have $$x= \pm 1 \implies p=\pm 2$$. Insufficient.

Combine (1) and (2), we have $$p= \pm 2$$. Insufficient.

To solve quickly in GMAT, just try to find two special values of $$p$$ are $$\pm 2$$. We will get the answer choice E.

The trick here is that $$p \in R$$ but we could consider value $$2$$ only.
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What does p equal   [#permalink] 02 Jul 2017, 20:39
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