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 OA is D I simplified the original equation to [spoiler]25z=3x^y[/spoiler] but couldnt deduce anything based on the statements. how can i logically figure this out?

hutch27 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 OA is D I simplified the original equation to [spoiler]25z=3x^y[/spoiler] but couldnt deduce anything based on the statements. how can i logically figure this out? (3²�)(35¹�)(z) = (5�)(7¹�)(9¹�)(x^y) (3²�)(7¹�5¹�)(z) = (5�)(7¹�)(3²)¹�(x^y) (3²�)(7¹�5¹�)(z) = (5�)(7¹�)(3²�)(x^y) (5²)(z) = (3)(x^y) z = (3) * (x^y)/5². The equation above implies that z is a multiple of 3 and that x^y is a multiple of 5². Statement 1: z is prime Since z is prime and a multiple of 3, z=3. Thus, (x^y)/5² = 1, implying that x=5 and y=2. SUFFICIENT. Statement 2: x is prime Since x^y is a multiple of 5² and x is prime, x=5 and y≥2. SUFFICIENT. The correct answer is D.

When you have variables as exponents (and all the variables are integers), you just need to ask yourself - what would allow one side to match the other? You simplified the equation to 25z=3x^y, which is exactly right. Now ask - what would allow both sides to be equal? - z has to have at least a 3 in it, because there's no 3 in 25 - x has to have at least a 5 in it, because it needs to account for the 25 - we don't know yet if z and x have other factors in them - 2's or 7's, etc. If x has other stuff in it, then z will have to match that stuff. So, in order to figure out what x is, we'd need to know know what restrictions there are on the possibilities. 1) z is prime Since we know that z has to have at least a 3, then that must be the prime - z doesn't have anything else in it. And if z is 3, then 25*3 = x^y, so x must be 5 and y must be 2. Sufficient. 2) x is prime We know that x has to have at least a 5, so x must equal 5. Sufficient.