Q.
If x, y, and z are integers greater than 1, and (3^27)(5^10)(z) = (5^8)(9^14)(x^y), then what is the value of x?

(1) y is prime

(2) x is prime


Answer from CAT:
The best way to answer this question is to use the rules of exponents to simplify the question stem, then analyze each statement based on the simplified equation.

(3^27)(5^10)(z) = (5^8)(9^14)(x^y) Simplify the 9^14
(3^27)(5^10)(z) = (5^8)(3^28)(x^y) Divide both sides by common terms 5^8, 3^27
(5^2)(z) = 3x^y

(1) INSUFFICIENT: Analyzing the simplified equation above, we can conclude that z must have a factor of 3 to balance the 3 on the right side of the equation. Similarly, x must have at least one factor of 5. Statement (1) says that y is prime, which does no tell us how many fives are contained in x and z.

For example, it is possible that x = 5, y = 2, and z = 3:
52 · 3 = 3 · 52

It is also possible that x = 25, y = 2, and z = 75:
52 · 75 = 3 · 252
52 · 52 · 3 = 3 · 252

(2) SUFFICIENT: Analyzing the simplified equation above, we can conclude that x must have a factor of 5 to balance out the 52 on the left side. Since statement (2) says that x is prime, x cannot have any other factors, so x = 5. Therefore statement (2) is sufficient.

The correct answer is B.

----The way i read the simplified equation is that x must be a 5 and y has to be 2, while z must contain one factor of 3. Answer explanation, provides a different point of view. Please advise.

Never mind. I figured it out. Disregard. Thanks

Glad to hear it!

_________________
Tim Sanders
Manhattan GMAT Instructor

Sorry but I don't really understand this problem. In (b) even if x is a prime, x can be 5^3 which is 125.
Therefore
5^2 * z = 5^3 * 3
So, 25 * z = 125 * 3

Which means that z can be 15 and the equation still balances out. The solution should be E!

Never mind, all we care about is the value of x, not z...

:-)

_________________
Jamie Nelson
ManhattanGMAT Instructor