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) = 36So, the answer to the target question is
xy = 36Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: x = 9Take
(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) = 45Case 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) = 36Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Answer: A
Cheers,
Brent