If x and y are positive integers and 5^x

Bunuel, I have a question, how did you know that you had to reorganize the equation in this way?

From \(5^x-5^y=2^{y-1}*5^{x-1}\) TO \(5^x-2^{y-1}*5^{x-1}=5^y\)

Thanks!

Not directed at me, However you can re-arrange it another way.

We have \(5^x-5^y = 2^{y-1}*5^{x-1}\), Dividing on both sides by \(5^{x-1}\), we have
\(5-5^{y+1-x} = 2^{y-1} \to 5 = 5^{y+1-x} + 2^{y-1}\). Now, as x and y are positive integers, the only value which \(2^{y-1}\) can take is 4.Thus, y-1 = 2, y = 3. Again, the value of \(5^{y+1-x}\) has to be 1, thus, y+1-x = 0 \(\to\) x = y+1 = 3+1 = 4. Thus, x*y = 4*3 = 12.

Thus, I think you could re-arrange in any order, as long as you get a tangible logic.

Faster: First constrain the possible answers. We know \(x>y\) since if \(x=y\) then the left-hand side is 0 or if \(x<y\) then the LHS is negative... but the RHS is always positive. Now act: divide both sides by \(5^{x-1}\) to get \(5-5^{y-x+1} = 2^{y-1}\).

Since the new RHS is a power of two, the LHS must equal 1, 2, or 4. The only power of 5 that gets us one of those is \(5-5^0=5-1=4\). That means \(y=3\) and thus \(x=4\).

Oh, and BTW: The thread title is different than the equality in your post. (Title should have 5^{x-1}, not 5^x.)

@Karishma, thanks again.

@SizeTrader, appreciate the solution and that reasoning was excellent. Also, the reason the title says 5^x is because I reached the character limit for the title and it got cut off

by dividing both sides of the equation with 5^x we have 1- 5^(y-x) = 2^(y-1) * 5^(x-1-x)

1- 5^(y-x) = 2^(y-1) * 5^(-1)

5= 2^(y-1) + 5^(y-x+1)

now minimum value of 2^(y-1) =1, hence 2^(y-1) must be equal to 4 and 5^(y-x+1) must be equal to 1 for the R.H.S to become equal to L.H.S.

2^(y-1) = 4 for y=3
and 5^(y-x+1) =1 for x=4 (as y=3)

hence product of xy = 12

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

E

Protocol:Simplify expression

1. need to find X and Y but cant isolate X and Y directly so start by separating the bases:

divide both sides by (5^{x-1}

left and right side simplifies to 5-5^{y-x+1} = 2^{y-1}

2. after simplifying, analyze.

Right side must be positive integer ( y is at least 1 ) thus left side must be positive too.
==> Left side is 5 minus an expression so answer must be at max 4 and at minimum 1. Right side must also be equal to a power of 2. Thus 4 is only possible answer for left side. Thus Y = 3.

Note: From original expression it is clear that X> Y. However by the time you get to simplifying the expression, the possible number of answers is so constrained that this information isn't critical to arriving at a faster answer.

i did another logic is it right?

\((5^x)(1-(2^y-1)/5)=5^y\)

then \((5^x-y-1)(5-(2^y-1))=1\)

which means this must be \(x-y-1= 0\)
and \(5-(2^y-1) = 1\) means also \(2^y-1 = 4\) then \(y-1=2\)
then y=3 replace y in old equation we get x =4 then finally xy =12

(5^x)-(5^y)=(2^(y-1))*(5^(x-1))
(5^x)(1-5^y-x) = (2^(y-1))*(5^(x-1))
5(1-5^y-x) = (2^(y-1))
10(1-5^y-x) = (2^(y))

We note that (2^(y)) is always positive. which translates to (1-5^y-x) >0 or y-x<0

So, 10(5^y-x)(5^(x-y)-1) = (2^(y))
5^(y-x+1)*2*(5^(x-y)-1) =(2^(y))

Now, RHS is 5^0, which means y-x+1 = 0, x-y =1

inputting values in above,

5^(0)*2*(5^(1)-1) =(2^(y))
2*4 = (2^(y))
implies y =3
x= 4

xy = 12

Hi,
The explanations are great. However, I took a lot of time figuring out how to simplify the equation and get to some solution and then started exploring otherway around which worked faster for me..

Difference between any 2 powers of 5 would always yields an even number.,i.e., a multiple of 2. However, difference of only consecutive powers of 5 yields a number that is only multiple of 2 and 5. Check->(25-5), (125-25), (625-125). Also check(625-25), (625-5), etc. Then, I simply had to check the options which had consecutive integers as factors. Only 12 worked out with 3 and 4 as factors.

This is not a foolproof solution but just another way of thinking incase you feel trapped in a question.

Given: \((5^x)-(5^y)=(2^{y-1})*(5^{x-1})\)

Divide both sides by \(5^{x-1}\) to get: \(5^1 - 5^{y-x+1} = 2^{y-1}\)

Simplify: \(5 - 5^{y-x+1} = 2^{y-1}\)

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

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

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

Since \(5^{y-x+1}\) is always positive, we can see that \(5 - 5^{y-x+1}\) cannot be greater than 5

So, the only possible values of \(5 - 5^{y-x+1}\) are 1, 2 or 4

In other words, it must be the case that:
case a) \(5 - 5^{y-x+1} = 2^{y-1} = 1\)
case b) \(5 - 5^{y-x+1} = 2^{y-1} = 2\)
case c) \(5 - 5^{y-x+1} = 2^{y-1} = 4\)

Let's test all 3 options.

case a) \(5 - 5^{y-x+1} = 2^{y-1} = 1\)
This means y = 1 (so that the right side evaluates to 1)
The left side, \(5 - 5^{y-x+1} = 1\), when \(5^{y-x+1} = 4\). Since x and y are positive integers, it's IMPOSSIBLE for \(5^{y-x+1}\) to equal 4
So, we can eliminate case a

case b) \(5 - 5^{y-x+1} = 2^{y-1} = 2\)
This means y = 2 (so that the right side evaluates to 2)
The left side, \(5 - 5^{y-x+1} = 2\), when \(5^{y-x+1} = 3\). Since x and y are positive integers, it's IMPOSSIBLE for \(5^{y-x+1}\) to equal 3
So, we can eliminate case b

case c) \(5 - 5^{y-x+1} = 2^{y-1} = 4\)
This means y = 3 (so that the right side evaluates to 4)
The left side, \(5 - 5^{y-x+1} = 4\), when \(5^{y-x+1} = 1\).
If \(5^{y-x+1} = 1\), then \(y-x+1 = 0\)
In this case, y = 3
So, we can write: 3 - x + 1 = 0, which mean x = 4
So, the only possible solution is y = 3 and x = 4, which means xy = (4)(3) = 12

Cheers,
Brent

\((5^x)-(5^y)=(2^{y-1})*(5^{x-1})\)
=\((5^{x-1})(5-5^{y-x+1})=(2^{y-1})*(5^{x-1})\)

5^{x-1} =/=0. So, \((5-5^{y-x+1})=(2^{y-1})\)
Since \((2^{y-1})\) must be +ve. Also Y is +ve so, y-1>0 and hence \((2^{y-1})\) will be an integer only.
Hence y-x+1 can be 0 only.
y-x+1 = 0
-> y-x = -1

also at y-x+1 = 0
\(5-1 = (2^{y-1})\)

\(4 = (2^{y-1})\)
y -1 = 2
y = 3

x= y+1 = 4
xy = 12


Answer E

Hi All,

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:

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

Good Question.. I solved it this way..

Let \(x-1 = a\), \(y-1 = b\).. So the equation becomes,

\(5^a - 5^b\) = \(\frac{5^a*2^b}{5}..\)
Now, we have to find the value of \(xy\) i.e. \((a+1)(b+1)\)
By looking at the options and the above equation only E satisfies.

We can simplify the given equation by dividing both sides by 5^(x - 1):

5^x/5^(x - 1) - 5^y/5^(x - 1) = 2^(y-1)

5 - 5^(y - x + 1) = 2^(y - 1)

It’s not easy to solve an equation with two variables by algebraic means; however, since both variables are positive integers, we can try numbers for one of the variables and solve for the other.

If we let y = 1, then the right hand side (RHS) = 2^0 = 1 and thus the left hand side (LHS) is 5^(1 - x + 1) = 4. However, a power of 5 can’t be equal to 4 when the exponent is an integer.

Now, let’s let y = 2; then the RHS = 2^1 = 2 and thus, for the LHS, 5^(2 - x + 1) = 3. However, a power of 5 can’t be equal to 3 when the exponent is an integer.

Finally, let’s let y = 3; then the RHS = 2^2 = 4 and thus, for the LHS, 5^(3 - x + 1) = 1. We see that a power of 5 can be equal to 1 when the exponent is 0. Thus:

3 - x + 1 = 0

x = 4

We see that x = 4 and y = 3 and thus xy = 12.

Answer: E

-- The simplest solution so far --

1/ Move the 5^(x-1) to the left side
2/ Move (1/2) from the right side 2^(y-1) = (2^y)*(1/2) to the left side
3/ Now you have 10 - 2*5^(y-x+1) = 2^y
4/ Ask when this is possible? -- The right side is always >0 so the left side must be >0 and this is true only if (y-x+1)=0
5/ Now you are looking for a combination of numbers from the choices that is y+1=x
6/ 3*4=12 looks good!

\(5^x-5^y=2^{y-1}*5^{x-1}\)

\(\frac{5^x}{5^{x-1}}-\frac{5^y}{5^{x-1}}=2^{y-1}\)

\(5-5^{y-x+1}=2^{y-1}\)

The blue equation implies the following:
5 - POWER OF 5 = POWER OF 2.
The only logical option is as follows:
\(5 - 5^0 = 2^2\).

Since the right side of the blue equation is equal to \(2^2\), we get:
\(2^{y-1}=2^2\)
\(y-1=2\)
\(y=3\).

Since y=3 and the subtracted term on the left side is equal to \(5^0\), we get:
\(5^{3-x+1}=5^0\)
\(3-x+1=0\)
\(4=x\).

Thus:
\(xy = 4*3 = 12\).

Clearly,
5^x - 5^y = 2^y-1 * 5^x-1
(5^x-5^y)/5^(x-1) = 2^(y-1)
5 - 5^(y-x+1) = 2^ (y-1)

Now,keeping conditions RHS always +ve .So LHS must always be a positive.
RHS , a factor of 2 can be made on the LHS by when 5^(y-x+1) = 1
So y-x+1 = 0
y-x=-1

Also y-1 =2
so y =3
x =4

which gives xy = 12.