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# If x,y are numbers >0 and 2^x*5^y=400

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If x,y are numbers >0 and 2^x*5^y=400 [#permalink]

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04 May 2013, 13:42
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If $$x,y$$ are numbers $$>0$$ and $$2^x*5^y=400$$, what could be the value of $$x$$?

I)2
II)3
III)4

A)Only I
B)Only II
C)I and II
D)Only III
E)I, II and III

Hi guys, this is my first PS question hope you like it, any feedback is appreciated.

(BTW: it's not as easy as it seems, it mays tricks you...)
[Reveal] Spoiler: OA

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Re: If x,y are numbers >0 and 2^x*5^y=400 [#permalink]

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04 May 2013, 18:32
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Zarrolou wrote:
If $$x,y$$ are numbers $$>0$$ and $$2^x*5^y=400$$, what could be the value of $$x$$?

I)2
II)3
III)4

A)Only I
B)Only II
C)I and II
D)Only III
E)I, II and III

Hi guys, this is my first PS question hope you like it, any feedback is appreciated.

I'll post the OA and the OE after some discussion. Kudos to the first ones who got it right!
(BTW: it's not as easy as it seems, it mays tricks you...)

I am getting E

You can make it into \sqrt{2^x} * \sqrt{5^y} =20

given that all numbers are >0

Therefor x=2 x^2*5^1=20 2 works

and when x=4 gives you 16*25=400

if x=3 it comes out to 2\sqrt{2}*5\sqrt{2}=400
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Re: If x,y are numbers >0 and 2^x*5^y=400 [#permalink]

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04 May 2013, 21:42
Zarrolou wrote:
If $$x,y$$ are numbers $$>0$$ and $$2^x*5^y=400$$, what could be the value of $$x$$?

I)2
II)3
III)4

A)Only I
B)Only II
C)I and II
D)Only III
E)I, II and III

Hi guys, this is my first PS question hope you like it, any feedback is appreciated.

I'll post the OA and the OE after some discussion. Kudos to the first ones who got it right!
(BTW: it's not as easy as it seems, it mays tricks you...)

First we represent 400 in its prime factored form.
$$400 = 4 * 100 = 2^2 * 2^2 * 5^2 = 2^4 * 5^2$$
Hence $$x = 4$$ is the only possible value.
Correct option is D
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Re: If x,y are numbers >0 and 2^x*5^y=400 [#permalink]

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04 May 2013, 22:48
Zarrolou wrote:
If $$x,y$$ are numbers $$>0$$ and $$2^x*5^y=400$$, what could be the value of $$x$$?

I)2
II)3
III)4

A)Only I
B)Only II
C)I and II
D)Only III
E)I, II and III

$$2^x*5^y=400$$
Substituting x=2,3,4 in the given equation
I) 2 --> $$2^2*5^y=400$$; $$5^y=100$$ --> Not possible
II) 3 --> $$2^3*5^y=400$$; $$5^y=50$$ --> Not possible
III) 4 --> $$2^4*5^y=400$$; $$5^y=25$$ , so y =2--> Correct

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Re: If x,y are numbers >0 and 2^x*5^y=400 [#permalink]

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05 May 2013, 00:15
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Official Explanation

If x,y are numbers >0 and $$2^x*5^y=400$$, what could be the value of x?

I)2
II)3
III)4

Read the question carefully! I said numbers not integers

$$2^x*5^y=2^4*5^2$$

III is true for sure. But pay attention that y (because could be a non integer) makes all the possible answers correct:

I)$$2^2*5^y=2^4*5^2$$, $$5^y=2^2*5^2$$, $$5^y=100$$, so y is a number $$2<y<3$$
II)$$2^3*5^y=2^4*5^2$$, $$5^y=2^1*5^2$$, $$5^y=50$$ , $$2<y<3$$

The correct answer is E) I, II and III
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Re: If x,y are numbers >0 and 2^x*5^y=400 [#permalink]

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21 Jul 2015, 08:47
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Re: If x,y are numbers >0 and 2^x*5^y=400 [#permalink]

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21 Jul 2015, 08:57
$$2^x * 5^y = 400$$

(i) if $$x = 2$$, then $$5^y = \frac{400}{2^2} = 100$$. So, $$5^y = 100$$ => $$2 < y < 3$$.

(ii) if $$x = 3$$, then $$5^y = \frac{400}{2^3} = 50$$. So, $$5^y = 50$$ => $$2 < y < 3$$.

(iii) if $$x = 4$$, then $$5^y = \frac{400}{2^4} = 25$$, hence $$y = 2$$.

All three cases are possible. Hence Ans (E).
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Re: If x,y are numbers >0 and 2^x*5^y=400 [#permalink]

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26 Feb 2016, 19:32
i understand that y can be a non-integer
but who can give an exact answer when 2^2 or 2^3 * 5^y, where y is positive, is equal to 400?
please, 1 single example, how 5^y = 100 or 50.
what power y should be? what fraction? with exact steps..
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Re: If x,y are numbers >0 and 2^x*5^y=400 [#permalink]

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26 Feb 2016, 19:52
mvictor wrote:
i understand that y can be a non-integer
but who can give an exact answer when 2^2 or 2^3 * 5^y, where y is positive, is equal to 400?
please, 1 single example, how 5^y = 100 or 50.
what power y should be? what fraction? with exact steps..

A suggestion: when you ask a question, ask in such a way that shows that you want to learn something. Be humble about it.

The question statement should have told you that the answer must be all of them as you can definitely calculate some sort of a value (using logs, not required for GMAT!). You are not supposed to get the exact value. All you need is to make sure that you get a real value, something that you will surely get for the question asked. As for the range, you can clearly see that 5^2<100<5^3, thus 2<y<3

As for the academic part of this discussion, I will use logs to get the value:

5^y=100 ---> take logs to the base 10 on both the sides:

y log 5 = 2 log 10 , log 10 =1 and log 5 = log 10 - log 2 = 1-0.301 = 0.699.

Thus y = 2.86245 , giving you $$5^y \approx 100$$

Again, this question is not GMAT like but is a good practice to understand how far you can go in 'solving' a particular question.

Hope this helps.
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Re: If x,y are numbers >0 and 2^x*5^y=400   [#permalink] 26 Feb 2016, 19:52
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