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# Is 5^k less than 1,000?

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Joined: 02 Dec 2012
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Is 5^k less than 1,000?  [#permalink]

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26 Dec 2012, 04:41
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Is 5^k less than 1,000?

(1) $$5^{(k+1)} > 3,000$$

(2) $$5^{(k-1)} = 5^k - 500$$
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Joined: 02 Sep 2009
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Re: Is 5^k less than 1,000?  [#permalink]

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26 Dec 2012, 04:42
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Is 5^k less than 1,000?

Is $$5^k<1,000$$?

(1) 5^(k+1) > 3,000 --> $$5^k>600$$ --> if $$k=4$$ then the answer is YES: since $$600<(5^4=625)<1,000$$ but if $$k=10$$, for example, then the answer is NO. Not sufficient.

(2) 5^(k-1) = 5^k - 500 --> we can solve for k and get the single numerical value of it, hence this statement is sufficient. Just to illustrate: $$5^k-5^{k-1}=500$$ --> factor out $$5^{k-1}$$: $$5^{k-1}(5-1)=500$$ --> $$5^{k-1}=125$$ --> $$k-1=3$$ --> $$k=4$$. Sufficient.

Hope it's clear.
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Re: Is 5^k less than 1,000?  [#permalink]

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23 May 2014, 14:08
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The biggest take-away here should be that we don't need to solve the St-2. The moment you start solving St2, you have fallen for GMAT's classic time waster trap. See below.

Statement1: As 5^(k+1) > 3,000 --> k>4 and hence insufficient
Statement2: We dont need to solve the equation. Since this is an EQUATION (and not an inequality) with one variable, we will get the exact value of k and we will be able to answer the question one way or the other. SUFFICIENT.
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Re: Is 5^k less than 1,000?  [#permalink]

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06 Apr 2013, 23:12
Is 5^k less than 1,000?

(1) 5^(k+1) > 3,000

(2) 5^(k-1) = 5^k - 500

I would to ask why is this wrong
(1) 5^(k+1) > 3000

5^(k+1) > 5^5
Hence, k+1= 5 , k =4

If so, 5^4 is less than 1000. The answer should b sufficient for (1).
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Re: Is 5^k less than 1,000?  [#permalink]

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07 Apr 2013, 23:03
LMKong wrote:
Is 5^k less than 1,000?

(1) 5^(k+1) > 3,000

(2) 5^(k-1) = 5^k - 500

I would to ask why is this wrong
(1) 5^(k+1) > 3000

5^(k+1) > 5^5
Hence, k+1= 5 , k =4

If so, 5^4 is less than 1000. The answer should b sufficient for (1).

First of all 5^5=3,125>3,000, thus 5^(k+1) > 3000 does NOT necessarily mean that 5^(k+1) > 5^5.

Next, even if we had 5^(k+1) > 5^5 it still does not mean that k+1=5. It means that k+1>5 --> k>4.

Hope it's clear.
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Re: Is 5^k less than 1,000?  [#permalink]

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21 Nov 2013, 11:05
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Statement(1) : 5^(k+1) > 3000
The above inequality can be reduced to 5^k > 600. From this, we clearly know few possible values for k i.e., 4,5,6,..
Substituting these values in the inequality given in the question gives away both yes and no answers.
k = 4, 5^(4-1) < 1000
k = 5, 5^(5-1) < 1000
k = 6, 5^(6-1) > 1000
Hence statement(1) is not sufficient.

Statement(2): 5^(k-1) = 5^k - 500
Reducing the above inequality, 4/5 * 5^k = 500
So 5^k = 625 = 5^4. Clearly k = 4 and the original inequality is satisfied: 5^4 < 1000.
Hence statement(2) is sufficient.

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Re: Is 5^k less than 1,000?  [#permalink]

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16 Jan 2015, 13:08
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Hi All,

1) At NO POINT does it state that K has to be an integer.
2) It's clearly based on exponents, so some exponent rules/patterns MUST be involved.

We're asked if 5^K is < 1,000. This is a YES/NO question.

Fact 1: 5^(K+1) > 3,000

In this Fact, notice how the exponent (K+1) differs from the exponent in the question (K). There's an exponent rule that accounts for this difference.

As an example, consider...
5^2 = 25
5^3 = 125
Notice how 5^3 is "5 times" greater than 5^2? This difference occurs because the base is 5 and we're increasing the exponent by 1. It can also be used in reverse....

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

This is a standard rule about "dividing" exponents with the same base --> we SUBTRACT the exponents.

With Fact 1, we're dealing with 5^(K+1) and the question is dealing with 5^K. This means that DIVIDING 5^(K+1) by 5 will give us 5^K:

5^(K+1)/5^1 = 5^(K+1-1) = 5^K.

This is all meant to say that we can DIVIDE both sides of this inequality by 5, which gives us...

5^(K+1) > 3,000
5^K > 600

IF....
5^K = 601 then the answer to the question is YES
5^K = 1,001 then the answer to the question is NO
Fact 1 is INSUFFICIENT

Fact 2: 5^(K-1) = 5^K - 500

This is a 1 variable, 1 equation "system", so we CAN solve it (and there will only be 1 answer). Even if you did not know that, it's still easy enough to get to the solution.... Since most Test Takers are better at basic multiplication than they are at manipulating higher-level exponents, here's how you can "brute force" the solution:

5^1 = 5
5^2 = 25
5^3 = 125
5^4 = 625
5^5 = 3125

Find two consecutive powers of 5 that differ by 500 and you have the solution to the above equation.
5^4 - 5^3 = 625 - 125 = 500
Fact 2 is SUFFICIENT.

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Re: Is 5^k less than 1,000?  [#permalink]

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08 Mar 2016, 22:18
Here the question will be fairly easy if we consider it as an inequality expression rather than Exponents as never does it state that K is an integer..
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Re: Is 5^k less than 1,000?  [#permalink]

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13 Jan 2018, 12:18
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Top Contributor
Is 5^k less than 1,000?

(1) $$5^{(k+1)} > 3,000$$

(2) $$5^{(k-1)} = 5^k - 500$$

Target question: Is 5^k less than 1000?

Statement 1: 5^(k+1) > 3000
First notice that 5^(k+1) = (5^k)(5^1)
So, we can take 5^(k+1) > 3000 and divide both sides by 5 to get: 5^k > 600
There are several possible cases to consider. Here are two:
case a: 5^k = 601, in which case 5^k is less than 1000.
case b: 5^k = 1001, in which case 5^k is not less than 1000.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT.

Statement 2: 5^(k-1) = 5^k - 500
IMPORTANT: Notice that we're given an EQUATION, which means we can solve the equation to find the definitive value of k. If we can find the value of k, then we can instantly tell whether or not 5^k is less than 1000. So, it SEEMS that we can conclude that statement 2 is sufficient WITHOUT performing any calculations. HOWEVER, if it's the case that the equation yields 2 possible values of k, then it may be the case that one value of k is such that 5^k is less than 1000, and the other value of k is such that 5^k is greater than 1000. So, at this point, we need only determine whether or not the equation will yield 1 or 2 values of k.

Rearrange to get the k's on one side: (5^k) - 5^(k-1) = 500
Factor the left side: 5^(k-1)[5 - 1] = 500
Simplify: 5^(k-1)[4] = 500
STOP!!
At this point, we can see that this equation will yield only one value of k. So, IF WE WERE to solve the equation for k, we would definitely be able to determine whether or not 5^k is less than 1000.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT.

Cheers,
Brent
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Re: Is 5^k less than 1,000?  [#permalink]

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13 Aug 2018, 13:05
Hi Brent GMATPrepNow and Bunuel,
Will this approach work

5^k< (5^3* 2^3)
So if K<= 4 then we can get answer for our question.

1) 5^(k+1)>3,000
5^(k+1)>3 * 5^3 * 2^3

so k+1>3 so k> 2 So insufficient.

5^(k−1)=5^k−500
5^k−5^(k−1)=500
5^k{ 1-1/5}= 500
5^k{4/5}=500
4*5^k= 2500
4*5^k=5^4 * 4
so k=4
So sufficient .
Re: Is 5^k less than 1,000? &nbs [#permalink] 13 Aug 2018, 13:05
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