Lesson: Equations and Powers

Comment on Equations and Powers

I need to understand the logic of this problem: How do we know what "x" equals and what "y" equals?

Here is what I am talking about:

OBSERVE: Notice that the right side, 2^(y−1), is POSITIVE for all values of y -----OK

Since y is a positive integer, 2^(y−1) can equal 1, 2, 4, 8, 16 etc (powers of 2----OK

So, the left side, 5 − 5^(y−x+1), must be equal 1, 2, 4, 8, 16 etc (powers of 2).-----OK

Since 5^(y−x+1) is always positive, we can see that 5 − 5^(y−x+1) cannot be greater than 5 ----so what are the values for x and y that I plug in ?

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
gmat-admin's picture

You're referring to my solution to this question: https://gmatclub.com/forum/if-x-and-y-are-positive-integers-and-5-x-1303...

We're not plugging in; we're solving equations.

Once we make all of the above conclusions, we know that there are only 3 cases (as shown in my solution)

Take case a for example: 5 − 5^(y−x+1) = 2^(y−1) = 1
If 2^(y−1) = 1, then y = 1
If 5 − 5^(y−x+1) = 1, then 5^(y−x+1) = 4
Since it's IMPOSSIBLE for 5^(some integer) to equal 4, we can eliminate case a.

Then we move onto case b...etc

For each case, we're solving an equation. No plugging in necessary.

You're referring to my solution to this question: https://gmatclub.com/forum/if-x-and-y-are-positive-integers-and-5-x-1303...
We're not plugging in; we're solving equations.
Once we make all of the above conclusions, we know that there are only 3 cases (as shown in my solution)
Take case “a” for example: 5 − 5^(y−x+1) = 2^(y−1) = 1
If 2^(y−1) = 1, then y = 1

Since y is the first multiple of 2(and since it has to be positive) we plug it in 2^(1−1) = 20 =1
OK

If 5 − 5^(y−x+1) = 1, then 5^(y−x+1) = 4

1) 5 − 5^(y−x+1) = 1
2) − 5^(y−x+1) = -4
3) (-1)^( − 5^(y−x+1) = -4)
4) 5^(y−x+1) = 4 (5 to any power cannot equal 4, so no good?)

So now I am assuming we use the next power of 2 (which is 2) and plug it in?

1) 5 − 5^(y−x+1) = 2
2) − 5^(y−x+1) = -5 + (2)
3) (-1)( − 5^(y−x+1) = -3 )
4) 5^(y−x+1) = 3 (5 to any power cannot equal 3, so no good?)

On to the next power of 2 (which is 4) and plug it in?

1) 5 − 5^(y−x+1) = 4
2) − 5^(y−x+1) = -5 + (4)
3) (-1)( − 5^(y−x+1) = -1)
4) 5^(y−x+1) = 1 (5 to “0” can equal 4, so good, right?)

Did I approach this the right way?

I don’t know how you would do this problem in under 2 minutes???
gmat-admin's picture

Yes, that's perfect.
It's a super tough question to answer quickly.

Add a comment

Ask on Beat The GMAT

If you have any questions, ask them on the Beat The GMAT discussion forums. The average response time is typically less than 30 minutes.

Tweet about our course!

If you're enjoying our video course, help spread the word on Twitter.

Have a question about this video?

Post your question in the Comment section below, and we’ll answer it as fast as humanly possible.