RiyaR wrote:If x, y, and z are integers greater than 1, and (327)(3510)(z) = (58)(710)(914)(xy), then what is the value of x?

(1) z is prime

(2) x is prime

Are you sure you've entered this correctly? It doesn't appear to have a solution.

Let's start by simplifying the original equation:

327 * 3510 * z = 58 * 710 * 914 * xy

Now let's try to factor each number.

327 = 3 * 109
3510 = 10 * 351 = 2 * 5 * 3 * 117 = 2 * 5 * 3 * 3 * 39 = 2 * 5 * 3 * 3 * 3 * 13

So 327 * 3510 is really 2 * 3� * 5 * 13 * 109

58 = 2 * 29
710 = 71 * 2 * 5
914 = 2 * 457

So 58 * 710 * 914 is really 2³ * 5 * 29 * 71 * 457

Setting the two equations equal to each other, we have

2 * 3� * 5 * 13 * 109 * z = 2³ * 5 * 29 * 71 * 457 * xy

which reduces (barely) to

3� * 13 * 109 * z = 2² * 29 * 71 * 457 * xy

Unfortunately, S1 is impossible. If z is prime, the LHS and the RHS cannot be equal, since the LHS needs to have ALL of the prime factors on the RHS, so z must be a multiple of 4, 29, 71, AND 457. Since S1 cannot be evaluated, the question can't be answered.

Eureka!

You meant to type 3²� * 35¹� * z = 5� * 7¹� * 9¹� * xʸ. This is a SERIOUS violation of our policy on properly typing questions, but I'll answer it anyway.

3²� * 35¹� * z = 5� * 7¹� * 9¹� * xʸ

is really

3²� * (5*7)¹� * z = 5� * 7¹� * (3*3)¹� * xʸ

or

3²� * 5¹� * 7¹� * z = 5� * 7¹� * 3²� * xʸ

Dividing out common prime factors, we get

5² * z = 3 * xʸ

S1 is SUFFICIENT: since 25z = a multiple of 3, and z is prime, z MUST be 3.
S2 is SUFFICIENT: since 3 * xʸ = a multiple of 25, and x is prime, x MUST be 5.

The problem should read as follows:

If x, y, and z are integers greater than 1, and (3²�)(35¹�)(z) = (5�)(7¹�)(9¹�)(x^y), then what is the value of x?

(1) z is prime

(2) x is prime

(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.
Since z is an INTEGER, the resulting equation 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.