II wrote:If x, y, and z are integers greater than 1, and (3^27)(35^10)(z) = (5^8 )(7^10)(9^14)(x^y), then what is the value of x?
(1) z is prime
(2) x is prime
The solutions posted are good, but sure, I'll throw my 2 cents in.
As always, when faced with a complicated stem we want to simplify it. Here, we see different very big numbers on both sides, so let's reduce to primes:
(3^27)(5^10)(7^10)z = (5^8)(7^10)(3^28)(x^y)
[35^10 = (5*7)^10 = 5^10 * 7^10]
[9^14 = (3^2)^14 = 3^(2*14) = 3^28]
Since we're solving for x, let's move all the numbers to the 'z' side. After reducing the exponents, we get:
(5^2)(z)/(3) = x^y
(25/3)z = x^y
As pointed out, we know that x, y and z are all integers greater than 1; therefore, x^y is an integer. Since the right side is an integer, the left side must also be an integer. For (25/3)z to be an integer, z MUST be a multiple of 3.
So, to solve for x, we need more info about x, y and/or z.
(1) z is prime.
Well, if z is a multiple of 3 AND z is prime, we know that z=3.
We now know that x^y = 25*3/3 = 25
If x and y are integers, there are two solutions:
a) x = 25 and y = 1; and
b) x = 5 and y = 2.
Since we know that y is greater than 1, x MUST be equal to 5. Sufficient!
(2) x is prime.
We could have written the original equation as:
z = (3/25)x^y
Since z is an integer, the right side must also be an integer. For the right side to be an integer, x^y has to be a multiple of 25.
For x^y to be a multiple of 25, x MUST be a multiple of 5 (since x and y are integers). If x is a multiple of 5 AND prime, x MUST be equal to 5. Sufficient!
Each statement is sufficient on its own: choose (d).
