Problem Solving

PauloAH wrote:If x and y are positive integers and 5^x - 5^y = [2^(y-1)][5^(x-1)], what is the value of xy?

A) 48
B) 36
C) 24
D) 18
E) 12

First notice that the right hand side of this equation will always be POSITIVE for any values of x and y.
So, we can conclude that the left side must be POSITIVE
In other words, 5^x - 5^y > 0
This means that x > y

If x > y, we can factor out 5^y from the left side, to get:
(5^y)[5^(x-y) - 1] = [5^(x-1)][2^(y-1)]
Aside: at this point, we can see that [5^(x-y) - 1] must evaluate to be some power of 2.
More importantly, we can see that 5^y = [5^(x-1)]
This tells us that y = (x-1)
In other words, x is 1 greater than y
At this point, we can solve the question without performing any more calculations. Here's why:
When we check the answer choices, ONLY ONE of them can be written as the product of 2 positive integers (x and y), where x is 1 greater than y
Only E (12) works here. We can write 12 as (4)(3)
So, it must be the case that x = 4 and y = 3

Let's check:
If x = 4 and y = 3, our original equation becomes: 5^4 - 5^3 = [2^(3-1)][5^(4-1)]
Simplify: 625 - 125 = [4][125]
Evaluate: 500 = 500...perfect!

Answer: E


Cheers,
Brent

Hi PauloAH,

GMAT Quant questions can almost always be solved in a variety of ways, so if you find yourself not able to solve by doing complex-looking math, then you should look for other ways to get to the answer. Think about what's in the question; think about how the rules of math "work." This question is LOADED with Number Property clues - when combined with a bit of "brute force", you can answer this question by doing a lot of little steps.

Here are the Number Properties (and the deductions you can make as you work through them):

1) We're told that X and Y are POSITIVE INTEGERS, which is a great "restriction."
2) We can calculate powers of 5 and powers of 2 rather easily:

5^0 = 1
5^1 = 5
5^2 = 25
5^3 = 125
5^4 = 625
Etc.

2^0 = 1
2^1 = 2
2^2 = 4
Etc.

3) The answer choices are ALL multiples of 3. Since we're asked for the value of XY, either X or Y (or both) MUST be a multiple of 3.

4)
*The left side of the equation is a positive number MINUS a positive number.
*The right side is the PRODUCT of two positive numbers, which is POSITIVE.
*This means that 5^X > 5^Y, so X > Y.

5)
*Notice how we have 5^X (on the left side) and 5^(X-1) on the right side; these are consecutive powers of 5, so the first is 5 TIMES bigger than the second.
*On the left, we're subtracting a number from 5^X.
*On the right, we're multiplying 2^(Y-1) times 5^(X-1).
*2^(Y-1) has to equal 1, 2 or 4, since if it were any bigger, then multiplying by that value would make the right side of the equation TOO BIG (you'd have a product that was bigger than 5^X).
*By extension, Y MUST equal 1, 2 or 3. It CANNOT be anything else.

6) Remember that at least one of the variables had to be a multiple of 3. What if Y = 3? Let's see what happens....

5^X - 5^3 = 2^2(5^(X-1))

5^X - 125 = 4(5^(X-1))

Remember that X > Y, so what if X = 4?.....

5^4 - 125 = 500
4(5^3) = 500

The values MATCH. This means Y = 3 and X = 4. XY = 12

Final Answer: E

GMAT assassins aren't born, they're made,
Rich

saadishah wrote:
Second conclusion which is not clear is that how come x is 1+y or 1 higher than y.

Are you okay with how we reached the conclusion that 5^y = [5^(x-1)]?
This tells us that y = (x-1)
We can rewrite this as y + 1 = x
In other words, x is 1 greater than y

Does that help?

Cheers,
Brent

saadishah wrote:What if x = 1 and y = 12.

If you plug x = 1 and y = 12 into the original equation, 5^x - 5^y = [2^(y-1)][5^(x-1)], it does not work out, so it CANNOT be the case that x = 1 and y = 12

Cheers,
Brent

Amrabdelnaby wrote:Hi Brent,

What's the level of this question?

I feel it is one of the top ones!

You're right - this is an incredibly difficult question! It comes from the Manhattan Prep Advanced Quant book. This book is designed for student who have already mastered the content and are scoring around a 700, but want to push themselves to score in the 750+ range.

As such, a lot of the questions you'll find here are harder than ones you'll see on the actual test. I describe them as "900-level" questions! The idea is, if you can understand the structures and patterns in these incredibly difficult questions, the real test won't feel so hard (we hope)!

osama_salah wrote:Sorry for this out-of-date reply but I have an irritating question: How long is solving this question supposed to take?

Like many of the questions in the AQ Guide, this one would take many test-takers (even expert-level ones) more than 2 minutes to solve. The questions in the AQ Guide are not meant to be a representative sample of GMAT questions. They're meant to simulate the hardest possible questions that someone might see if she/he was scoring near a 51 on quant.

My recommendation - don't worry about timing yourself on AQ problems. Just use them for mental muscle-building.