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.

Hi Brent, the answer given to the question below is c. I am failing to understand why?
https://gmatclub.com/forum/is-4-x-y-139120.html

Is 4^(x+y) = 8^10?

(1) x - y = 9
(2) y/x = 1/4

The equation becomes: x+y = 15

We can solve both the equation with both the statements. Therefore both are sufficient for me.
gmat-admin's picture

Be careful. You have accidentally turned the target QUESTION into a true STATEMENT.

The target question asks "Is 4^(x+y) = 8^10?"
We can rephrase this to get: "Does x + y = 15?"

We still have a QUESTION on our hands. We don't know whether or not x + y = 15. Our goal is to determine whether each statement provides enough information to answer that question.

Statement 1) x - y = 9
Does this statement provide sufficient information to answer the rephrased target question ("Does x + y = 15?")?
No.
If x - y = 9, then it's possible that x = 12 and y = 3, in which case the answer to our rephrased target question is "YES, x + y does equal 15
If x - y = 9, then it's also possible that x = 10 and y = 1, in which case the answer to our rephrased target question is "NO, x + y does NOT equal 15

So, statement 1 is not sufficient.

Does that help?

By the way, here's my step-by-step solution: https://gmatclub.com/forum/is-4-x-y-139120-20.html#p1901036

Cheers,
Brent

Hi Brent, could you please help me with an explanation to this question below?

https://gmatclub.com/forum/x-is-an-integer-and-x-raised-to-any-odd-integer-is-greater-95476.html

https://gmatclub.com/forum/x-is-an-integer-and-x-raised-to-any-odd-integer-is-greater-95476.html

can you please help me with an explanation.
gmat-admin's picture

You bet.
Here's my full solution: https://gmatclub.com/forum/x-is-an-integer-and-x-raised-to-any-odd-integ...

Cheers,
Brent

Can you explain this clearly? Looks very confusing.
x is an integer and x raised to any odd integer is greater than zero; is w - z greater than 5 times the quantity 7^(x-1)-5^x?

(1) z < 25 and w=7^x
(2) x = 4
gmat-admin's picture

I have answered most of the questions in the Reinforcement Activities boxes. So, be sure to check whether I've answered a certain question.

Here's my full solution: https://gmatclub.com/forum/x-is-an-integer-and-x-raised-to-any-odd-integ...

Cheers,
Brent

how to simplify the equation here
If x and y are positive integers and (5x)−(5y)=(2y−1)∗(5x−1)(5x)−(5y)=(2y−1)∗(5x−1), what is the value of xy?

A. 48
B. 36
C. 24
D. 18
E. 12
gmat-admin's picture

Hi Brent,

These questions (marked as difficulty: 650 to 800) are very tricky. Wondering if we actually get questions this hard in the GMAT?
gmat-admin's picture

From experience, I can tell you that the questions I've listed listed in the Reinforcement Activities are within the scope of the GMAT. That said, some of them are in the 750-800 range, which means most test-takers won't see those kinds of questions on test day. Keep in mind that the GMAT scoring algorithm tries to locate questions that are barely within or barely beyond your abilities.

Hi Brent, how do you determine which instances you would factor out an exponent itself vs an exponent +/- a number?

e.g. factor out exponent itself like the 15^x here:
https://gmatclub.com/forum/if-x-and-y-are-integers-and-15-x-15-x-1-4-y-15-y-wha-99354.html

e.g. factor out exponent +/- number like the 3^(a–2) here: https://gmatclub.com/forum/if-3-a-3-a-2-8-3-27-what-is-the-value-of-2a-249179.html

Thank you.
gmat-admin's picture

Great question!

In both cases I'm doing the exact same thing.
With each factorization, the greatest common factor of both terms is the term with the smaller exponent.
Take, for example, this expression: x^8 + x^3 - x^7
Since 3 is the smallest exponent, the greatest common factor of all three terms is x^3, which will factor out as follows:
x^8 + x^3 - x^7 = x^3(x^5 + 1 - x^4)

Now, let's examine both questions, starting with https://gmatclub.com/forum/if-x-and-y-are-integers-and-15-x-15-x-1-4-y-1...
Here, we have: 15^x + 15^(x+1)
The exponents are x and x+1
So 15^x is the term with the smallest exponent, which means our factorization looks like this:
15^x + 15^(x+1) = 15^x(1 + 15^1)

Now let's examine https://gmatclub.com/forum/if-3-a-3-a-2-8-3-27-what-is-the-value-of-2a-2...
Here, we have: 3^a – 3^(a–2)
The exponents are a and a-2
So 3^(a–2) is the term with the smallest exponent, which means our factorization looks like this:
3^a - 3^(a–2) = 3^(a–2)[3^2 - 1]

Does that help?

That makes sense, thank you so much!

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