If x and k are both integers, x k, and x−k = 625, what is x? (1) |k|

Question

If x and k are both integers, x > k, and  x^{−k} = 625 , what is x?

(1) |k| is a prime number 
(2) x + k > 20

ccooley · Accepted Answer

Interesting problem!

The first thing I noticed is that the question itself really limits the possibilities. There can't be that many integers where  x^{−k} = 625 , right? Also, I know 625 as a 'special number' - you should memorize the perfect squares up to about 25^2, so that you notice things like this quickly on test day. 625 = 25^2, so I immediately know one of the possibilities. x could be 25, and k could be -2. (Note the 'double negative' there).

However, you should never be able to solve a DS question  without either statement . That's something that never happens on DS. So, there must be at least one possibility. The GMAT likes to trick you into forgetting about the simplest exponent of all: 1. x could be 625, and k could be -1.

Also, notice that 25 can be factored down more. 25^2 = 5^4. So, finally, x could be 5, and k could be -4.

List the three possibilities on your paper:
x = 25, k = -2
x = 625, k = -1
x = 5, k = -4

Then, start working with the statements. Your question to ask yourself: do the statements let me 'narrow it down' to just one of these possibilities?

(1) does exactly that. 2 is the only prime value for k in our list. So, if we know that |k| is prime, then the first possibility is the only one that works. (1) is sufficient.

(2) is insufficient, because the first two possibilities could both work. (They're really hoping that you don't think of x = 625, k=-1. If you didn't think of that, you'd think this was sufficient as well.)

tinayni552 · Answer

x can be a negative number.. so it can be equal to -5 no? which means that there are two options for the first statement.

Bunuel · Answer

Notice that we are given that x > k, so x cannot be -5 if k = -2.

tinayni552 · Answer

Thanks! I should be more careful when reading the stem

sandysilva · Answer

Hi Bunuel here how i saw statement (1) |k| is a prime number

But k can be -2 or 2. So when k = -2 and when k = 2 the equation  x^{−k} = 625  should give different results !
What am I missing here ?

Regards

Sandy da Silva

Bunuel · Answer

Not sure I understand what you mean. We have the following possible cases for  x^{−k} = 625 :

x = 25, k = -2
x = 625, k = -1
x = 5, k = -4

(1) says: |k| is a prime number, thus k = -2 and x = 25.

EMPOWERgmatRichC · Answer

Hi All,

In certain DS questions, the prompt significantly limits the possible answers (before you even consider the information in the two Facts). By determining those limited options right from the start, you'll find that the rest of the work needed to answer the question can be done rather quickly.

Here, we're told that X and K are both INTEGERS, that X > K and that X^{-K} = 625. We're asked for the value of X.

To start, there are not that many ways to raise an INTEGER to an INTEGER power and get 625; considering that X must be GREATER than K, it's even more limited - there are only 3 ways to do it:

X = 625, K = -1 
X = 25, K = -2 
X = 5, K = -4

1) |K| is a prime number

Given the above three options, there's only one option that 'fits' Fact 1: X = 25, K = -2 
Fact 1 is SUFFICIENT

2) X + K > 20

With Fact 2, there are two options (X = 25, K= -2 and X = 625, K = -1) 
Fact 2 is INSUFFICIENT

Final Answer:  A

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

Alexey1989x · Answer

case 1: x^k=25^-2
case 2: x^k=5^-4
case 3: x^k=625^-1
(1) |x| is a prime, so only case 1 applies ===> x=25
(2) case 1 & 3 apply: 
25-2=23 
625-1=624

Answer A.

Andrewcoleman · Answer

one thing that helped me initially keep the inequalities straight was to represent them like you would see on a number line which means the less than sign is always used to represent the number on the left is always less than the number on the right. after a while i didn't need it, but it was one less thing to keep track of mentally.

RaghavKhanna · Answer

x -> positive, negative
k -> positive, negative
x > k
x^−k=625
=> (1/x)^k = 625

St. 1:   
|k| -> prime number
=> |k| = 2,3,5,.... 
=> k = ... -5,-3,-2,2,3,5.....
if k = positive 
=> x cannot be an integer
=> k -> negative prime integer
if k = -2
=> x = 25
=>   St. 1 is sufficient

St. 2:   
x + k > 20
=> x = 25, k = -2
or, x = 625, k = -1

=>   St. 2 is insufficient

Answer: A

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If x and k are both integers, x > k, and x−k = 625, what is x? (1) |k|

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