Data Sufficiency

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 best way to answer this question is to use the exponential rules to simplify the question stem, then analyze each statement based on the simplified equation.

(3^27)(35^10)(z) = (5^8 )(7^10)(9^14)(x^y) Break up the 35^10 and simplify the 9^14
(3^27)(5^10)(7^10)(z) = (5^8 )(7^10)(3^28 )(x^y) Divide both sides by common terms 5^8, 7^10, 3^27
(5^2)(z) = 3x^y

(1) SUFFICIENT: 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. Statement (1) says that z is prime, so z cannot have another factor besides the 3. Therefore z = 3.

I am struggling to understand how (1) can be sufficient:

(5^2)(z) = 3x^y
How can we conclude that z MUST have a factor of 3 to balance the right side of the equation ? Why cant z have another factor besides 3 ?

Thanks in advance for your kind help.

II wrote:I am struggling to understand how (1) can be sufficient:

(5^2)(z) = 3x^y
How can we conclude that z MUST have a factor of 3 to balance the right side of the equation ? Why cant z have another factor besides 3 ?

Thanks in advance for your kind help.

Let's rewrite it:

z = (3*x^y)/(5^2)

z/3 = (x^y)/(5^2)

Now, we know that x^y is an integer (since x and y are integers) and we know that z is an integer. Given that we have 3 as the denominator on one side and 25 on the other, the only way that both sides can be equal is if each side reduces to an integer.

So, if we know that z MUST be an integer, that z MUST be a multiple of 3 and that z MUST be prime, we know that z=3.