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If x, y, and z are integers and 2^x*5^y*z = 6.4*10^6, what is the valu

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If x, y, and z are integers and 2^x*5^y*z = 6.4*10^6, what is the valu  [#permalink]

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New post 12 Jun 2015, 03:34
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66% (01:57) correct 34% (02:04) wrong based on 318 sessions

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Re: If x, y, and z are integers and 2^x*5^y*z = 6.4*10^6, what is the valu  [#permalink]

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New post 12 Jun 2015, 04:31
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Bunuel wrote:
If x, y, and z are integers and \(2^x*5^y*z = 6.4*10^6\), what is the value of xy?

(1) z = 20
(2) x = 9


Kudos for a correct solution.


CONCEPT: Such Questions always require Prime factorization on both side of the equation to compare the exponent of the same base values

Given: \(2^x*5^y*z = 6.4*10^6\)
i.e. \(2^x*5^y*z = 64*10^5\)
i.e. \(2^x*5^y*z = 2^{11}*5^5\)

Question : xy = ?

Statement 1: z = 20
i.e \(2^x*5^y*20 = 2^{11}*5^5\)
i.e \(2^{x+2}*5^{y+1} = 2^{11}*5^5\)
i.e. x+2 = 11 and y+1 = 5
i.e. x = 9 and y = 4

Hence, SUFFICIENT

Statement 2: x = 9
i.e \(2^9*5^y*z = 2^{11}*5^5\)
i.e. z must a multiple of \(2^2\) for the exponent of 2 to be equal on both sides of the equation
But y still remains unknown

Hence, NOT SUFFICIENT
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If x, y, and z are integers and 2^x*5^y*z = 6.4*10^6, what is the valu  [#permalink]

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New post Updated on: 15 Jun 2015, 08:31
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If x, y, and z are integers and \(2^x\)∗\(5^y\)∗z=6.4∗10^6, what is the value of xy?

(1) z = 20

\(2^x\)∗\(5^y\)∗20 = \(2^5\)*\(10^4\)*20
\(2^x\)*\(5^y\) = \(2^9\)*\(5^4\)

x=9,y=4
xy=36
Sufficient

(2) x = 9

\(2^9\)*\(5^y\)*z=\(2^6\)*\(10^5\)
\(2^9\)*\(5^y\)*z=\(2^{11}\)*\(5^5\)
if y=5 then z=4
if y=4 then z=20
if y=3 then z=100

So y can take any value ranging from 0 to 5.

Insufficient

Ans : A

Originally posted by ManojReddy on 12 Jun 2015, 04:54.
Last edited by ManojReddy on 15 Jun 2015, 08:31, edited 1 time in total.
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Re: If x, y, and z are integers and 2^x*5^y*z = 6.4*10^6, what is the valu  [#permalink]

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New post 15 Jun 2015, 05:42
Bunuel wrote:
If x, y, and z are integers and \(2^x*5^y*z = 6.4*10^6\), what is the value of xy?

(1) z = 20
(2) x = 9


Kudos for a correct solution.


MANHATTAN GMAT OFFICIAL SOLUTION:

Express both sides of the equation in terms of prime numbers. \(2^x*5^y*z = 6.4*10^6=2^{11}*5^5\).

The right side of the equation is composed of only 2's and 5's. The left side of the equation has x number of 2's and y number of 5's along with some factor z. This unknown factor z must be composed of only 2's and/or 5's, or it must be 1 (i.e. with no prime factors).

If z = 1, then x = 11 and y = 5.
If z = 2^?*5^?, where the exponents are not 0, then x and y will depend on the value of those exponents.

The rephrased question is thus “How many factors of 2 and 5 are in z?”

(1) SUFFICIENT: If \(z = 20 = 2^2*5^1\), then we have the answer to our rephrased question. Incidentally,this implies that \(2^x*5^y(2^2*5^1) = 2^{11}*5^5\), so x = 9 and y = 4, making xy = 36.

(2) INSUFFICIENT: If x = 9, then

\(2^x*5^y*z = 2^{11}*5^5\)

\(2^9*5^y*z = 2^{11}*5^5\)

\(z=2^2*5^{5-y}\)

While this tells us the number of 2's among z's factors, we still do not know how many factors of 5 are in z.

The correct answer is A.
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If x,y and z are integers and 2^x 5^y z = 6.4*10^6 , what is the value  [#permalink]

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New post 22 Apr 2018, 07:10
If x,y and z are integers and \(2^x 5^y z = 6.4*10^6\) , what is the value of xy?

1.z=20
2.x=9
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If x,y and z are integers and 2^x 5^y z = 6.4*10^6 , what is the value  [#permalink]

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New post 22 Apr 2018, 07:28
selim wrote:
If x,y and z are integers and \(2^x 5^y z = 6.4*10^6\) , what is the value of xy?

1.z=20
2.x=9


\(2^x 5^y z = 6.4*10^6=>64*10^5\)

\(2^x 5^y z = 2^{10}5^5\). Hence to know the value of \(xy\), we need to know the value of \(z\)

Statement 1: provides the value of \(z\). Sufficient

For the sake of calculation, \(2^x 5^y*20 = 2^{10}5^5=>2^{x+2}*5^{y+1}=2^{10}5^5\)

\(=>x+2=10=>x=8\)

\(=>y+1=5=>y=4\). Hence \(xy=8*4=32\)

Statement 2: nothing can be said about the value of y & z. Insufficient

Option A
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Re: If x, y, and z are integers and 2^x*5^y*z = 6.4*10^6, what is the valu  [#permalink]

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New post 22 Apr 2018, 21:47
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Re: If x, y, and z are integers and 2^x*5^y*z = 6.4*10^6, what is the valu  [#permalink]

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New post 09 May 2018, 00:01
I) 2^x*5^y=32*10^4

So y=4 and x=9
To make up equality of eqn so sufficient

II) we know x but we don't know z
Z=1 or z=5 or 10
Can result in different values of xy
So insufficient

A is answer

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Re: If x, y, and z are integers and 2^x*5^y*z = 6.4*10^6, what is the valu  [#permalink]

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New post 24 Jun 2018, 06:12
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Bunuel wrote:
If x, y, and z are integers and \(2^x*5^y*z = 6.4*10^6\), what is the value of xy?

(1) z = 20
(2) x = 9



Target question: What is the value of xy?

Given: x, y, and z are integers and (2^x)(5^y)(z) = (6.4)(10^6)
Since the left side of the given equation has been factored by primes, let's find the prime factorization of (6.4)(10^6)
(6.4)(10^6) = (6.4)(10^1)(10^5)
= (64)(10^5)
= (2^6)(10^5)
= (2^6)(2^5)(5^5)
= (2^11)(5^5)

So, we have: (2^x)(5^y)(z) = (2^11)(5^5)

Statement 1: z = 20
Take (2^x)(5^y)(z) = (2^11)(5^5) and replace z with 20...
We get: (2^x)(5^y)(20) = (2^11)(5^5)
Rewrite 20 as follows: (2^x)(5^y)(2^2)(5) = (2^11)(5^5)
Divide both sides of the equation by 2^2 to get: (2^x)(5^y)(5) = (2^9)(5^5)
Divide both sides of the equation by 5 to get: (2^x)(5^y) = (2^9)(5^4)
At this point, we can see that x = 9 and y = 4, so xy = (9)(4) = 36
So, the answer to the target question is xy = 36
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x = 9
Take (2^x)(5^y)(z) = (2^11)(5^5) and replace x with 9...
We get: (2^9)(5^y)(z) = (2^11)(5^5)
Divide both sides of the equation by 2^9 to get: (5^y)(z) = (2^2)(5^5)
From here, we can see that there are several values of x and z that satisfy the equation (5^y)(z) = (2^2)(5^5) .
Here are two possible cases:
Case a: y = 5 and z = 2^2. We already know that x = 9, so the answer to the target question is xy = (9)(5) = 45
Case b: y = 4 and z = (2^2)(5). We already know that x = 9, so the answer to the target question is xy = (9)(4) = 36
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

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Brent
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Re: If x, y, and z are integers and 2^x*5^y*z = 6.4*10^6, what is the valu  [#permalink]

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New post 26 Jun 2018, 18:11
Bunuel wrote:
If x, y, and z are integers and \(2^x*5^y*z = 6.4*10^6\), what is the value of xy?

(1) z = 20
(2) x = 9


Notice that 6.4 * 10^6= 64 * 10^5 = 2^6 * 2^5 * 5^5 = 2^11 * 5^5

Statement One Alone:

z = 20

Thus we have:

2^x + 5^y * 20 = 2^11 * 5^5

2^x + 5^y * 2^2 * 5 = 2^11 * 5^5

2^(x+2) + 5^(y+1) = 2^11 * 5^5

Since x and y are integers, x + 2 = 11 and y + 1 = 5. So x = 9 and y = 4 and xy = 9(4) = 36. Statement one alone is sufficient.

Statement Two Alone:

x = 9

Thus we have:

2^9 * 5^y * z = 2^11 * 5^5

5^y * z = 2^2 * 5^5

However, we can’t determine a unique value for y. For example, if z = 4, then y = 5. On the other hand, if z = 20, then y = 4. Statement two alone is not sufficient.

Answer: A
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Re: If x, y, and z are integers and 2^x*5^y*z = 6.4*10^6, what is the valu  [#permalink]

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New post 27 Jun 2018, 05:14
math revolution
how to apply variable approach here..??
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Re: If x, y, and z are integers and 2^x*5^y*z = 6.4*10^6, what is the valu &nbs [#permalink] 27 Jun 2018, 05:14
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