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

Question

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 .

GMATinsight · Accepted Answer

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

ManojReddy · Answer

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

Bunuel · Answer

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.

KSBGC · Answer

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

niks18 · Answer

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

Bunuel · Answer

Merging topics. Similar question to practice: https://gmatclub.com/forum/if-x-y-and-z ... 88182.html

push12345 · Answer

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

Posted from my mobile device

BrentGMATPrepNow · Answer

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

Cheers,
Brent

JeffTargetTestPrep · Answer

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

gmatify17 · Answer

math  revolution
how to apply variable approach here..??

bumpbot · Answer

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If x, y, and z are integers and 2^x5^yz = 6.4*10^6, what is the valu

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