Data Sufficiency

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

OA: B

@ Experts - sort of lost here. What is the best way to approach this kind of problem ? Please help!

RBBmba@2014 wrote: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

OA: B

@ Experts - sort of lost here. What is the best way to approach this kind of problem ? Please help!

(3²�)(5¹�)(z) = (5�)(9¹�)(x^y)

(3²�)(5¹�)(z) = (5�)(3²)¹�(x^y)

(3²�)(5¹�)(z) = (5�)(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: y is prime
Given that x^y is a multiple of 5² and y is prime, the following cases are possible:
x=5 and y=2
x=25 and y=2.
Since x can be different values, INSUFFICIENT.
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 B.

GMATGuruNY wrote: (5²)(z) = (3)(x^y)

z = (3) * (x^y)/5².

A quick question - is there any way to quickly understand under time constraint that taking z on the left side would be the best way to solve this problem ? (THIS COULD SAVE some good amount of time, i think!)

Or we just need to perform a TRIAL to best understand which variable should be placed on the left hand side of the equation, depending on the type of the equation/problem ?

A quick question - is there any way to quickly understand under time constraint that taking z on the left side would be the best way to solve this problem ? (THIS COULD SAVE some good amount of time, i think!)

Say you didn't think to isolate z. Once you simplify to:(3²�)(5¹�)(z) = (5�)(3²�)(x^y) we can see that because we have a 5^10 on the left side of the equation and a 5^8 on the right, we'll need at least an additional 5^2 on the right in order to have a balanced equation. So x^y must be at multiple of 5^2. Similarly, because 3^27 is on the left and 3^28 is on the right, we'll need an additional 3 on the left, so z will have to be a multiple of 3. When Mitch isolated z he was making this relationship clearer, but you shouldn't think that it's essential to isolate z in order to solve the problem in a reasonable amount of time.

DavidG@VeritasPrep wrote:

A quick question - is there any way to quickly understand under time constraint that taking z on the left side would be the best way to solve this problem ? (THIS COULD SAVE some good amount of time, i think!)

Say you didn't think to isolate z. Once you simplify to:(3²�)(5¹�)(z) = (5�)(3²�)(x^y) we can see that because we have a 5^10 on the left side of the equation and a 5^8 on the right, we'll need at least an additional 5^2 on the right in order to have a balanced equation. So x^y must be at multiple of 5^2. Similarly, because 3^27 is on the left and 3^28 is on the right, we'll need an additional 3 on the left, so z will have to be a multiple of 3. When Mitch isolated z he was making this relationship clearer, but you shouldn't think that it's essential to isolate z in order to solve the problem in a reasonable amount of time.

Alright. So, even if we take x^y on the left side then also it comes down to the fact that "x^y will be a multiple of 5²" as follows -

x^y = (z/3) * 5²
[As x^y has to be an INTEGER, z/3 MUST yield an integer value. So,z MUST be a multiple of 3 and subsequently this results in x^y to be a multiple of 5²]

Correct me please if my understanding is wrong!

RBBmba@2014 wrote:

GMATGuruNY wrote: (5²)(z) = (3)(x^y)

z = (3) * (x^y)/5².

A quick question - is there any way to quickly understand under time constraint that taking z on the left side would be the best way to solve this problem ? (THIS COULD SAVE some good amount of time, i think!)

Or we just need to perform a TRIAL to best understand which variable should be placed on the left hand side of the equation, depending on the type of the equation/problem ?

When given an equation with more than one variable on the lefthand side, I typically rephrase the equation so that one variable is in terms of the others.
Given equation: x - y - z = 2.
Rephrased equation: x = y + z + 2.

It is difficult to rephrase (5²)(z) = (3)(x^y) in terms of x or y.
It is far easier to isolate z:
z = (3) * (x^y)/5².

x^y = (z/3) * 5²
[As x^y has to be an INTEGER, z/3 MUST yield an integer value. So,z MUST be a multiple of 3 and subsequently this results in x^y to be a multiple of 5²]

Yep! Many paths, same destination.

(But I do agree with Mitch that it's easier to isolate z.)