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

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.

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

Merging topics.

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

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

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

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

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

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