Is 5^k less than 1,000? : GMAT Data Sufficiency (DS)
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# Is 5^k less than 1,000?

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Is 5^k less than 1,000? [#permalink]

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07 Dec 2010, 17:19
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Is 5^k less than 1,000?

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

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

OPEN DISCUSSION OF THIS QUESTION IS HERE: is-5-k-less-than-1-000-1-5-k-128055.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 23 May 2014, 01:02, edited 2 times in total.
Renamed the topic, edited the question and added the OA.
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Re: Variable in Exponent DS problem [#permalink]

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07 Dec 2010, 17:50
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tonebeeze wrote:
I would appreciate it if someone could walk me through this problem. Thanks

Is $$5^k$$ less than 1000?

1. $$5^k+1 > 3000$$
2. $$5^k-1 = 5^k -500$$

I am assuming the question is:

Is $$5^k$$ less than 1000?

1. $$5^{k+1} > 3000$$
2. $$5^{k-1} = 5^k -500$$

$$5^4 = 625$$ and $$5^5 = 3125$$ (even if you do not know this, it is fine. You don't need to calculate. Just observe that 625*5 will be greater than 3000)

Statement 1: $$5^{k+1} > 3000$$
This means k + 1 is greater than 4 so k is greater than 3 (It doesnt mean that k + 1 is at least 5 because the question doesn't say that k is an integer. k + 1 could be 4.999 making k = 3.999) Since k can take values less than 4 and more than 4, 5^k could be less than 1000 or more than 1000. Not sufficient.

Statement 2: $$5^{k-1} = 5^k -500$$
Re-arrange: $$500 = 5^k -5^{k-1}$$
$$5^3 *4 = 5^{k-1}(5 - 1)$$
Hence k - 1 = 3 and k = 4
So $$5^k$$ = 625 which is less than 1000. Answer is 'Yes'. Sufficient.

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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Manager Joined: 13 Jul 2010 Posts: 169 Followers: 1 Kudos [?]: 73 [0], given: 7 Re: Variable in Exponent DS problem [#permalink] ### Show Tags 07 Dec 2010, 18:18 Great point that (K+1) in stmt 1 does not mean more than 4 or at least 5, we forget non-integers when considering DS questions. Thanks. Intern Joined: 19 Dec 2010 Posts: 28 Followers: 0 Kudos [?]: 17 [0], given: 4 Re: Variable in Exponent DS problem [#permalink] ### Show Tags 02 Jan 2011, 19:03 Can you go over how you went from 500 = 5^k -5^{k-1} to 5^3 *4 = 5^{k-1}(5 - 1)? Thanks so much in advance! Manager Status: Current MBA Student Joined: 19 Nov 2009 Posts: 127 Concentration: Finance, General Management GMAT 1: 720 Q49 V40 Followers: 13 Kudos [?]: 350 [0], given: 210 Re: Variable in Exponent DS problem [#permalink] ### Show Tags 02 Jan 2011, 19:30 m990540 wrote: Can you go over how you went from 500 = 5^k -5^{k-1} to 5^3 *4 = 5^{k-1}(5 - 1)? Thanks so much in advance! Please refer to this link. It will help you easily post mathematic symbols in your posts. writing-mathematical-symbols-in-posts-72468.html Regarding the problem: $$500 = 5^k - 5^{k-1}$$ 500 can be simplified into $$125$$ x $$4$$, which is equal to $$5^3$$ x $$4$$ Step 1: Recognize that $$5^k$$ is a common factor in both $$5^k$$ and $$5^{k -1}$$. Proceed to factor out $$5^k$$ Step 2: $$5^k (1 - 5^{-1}) =500$$ ---> $$5^k (1 - \frac {1}{5}) = 500$$ ---> $$5k (\frac {4}{5}) = 500$$ --->$$5^k = \frac {5}{4} (500)$$ ---> $$5^k = 62$$5 Hope this helps Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7125 Location: Pune, India Followers: 2138 Kudos [?]: 13694 [0], given: 222 Re: Variable in Exponent DS problem [#permalink] ### Show Tags 03 Jan 2011, 07:29 Expert's post 1 This post was BOOKMARKED m990540 wrote: Can you go over how you went from 500 = 5^k -5^{k-1} to 5^3 *4 = 5^{k-1}(5 - 1)? Thanks so much in advance! Left hand side: $$500 = 5*100 = 5*25*4 = 5^3*4$$ (Since your concern is the power of 5, separate 5s from the rest) Right hand side: $$5^k -5^{k-1} = 5^{k - 1} ( 5 - 1)$$(Take $$5^{k - 1}$$ common. e.g. if you have $$5^4 - 5^3$$, you can take $$5^3$$ common and you will be left with (5 - 1)) _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: Variable in Exponent DS problem [#permalink]

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03 Jan 2011, 08:14
5^k+1 > 3000
5. 5^k > 3000
5^k > 600 (This is not sufficient to tell that it is less than 1000)
5^k-1 = 5^k -500
5^k / 5 = 5^k – 500
5^k = 5^k+1 – 2500
5^k (5 -1) = 2500
5^k = 2500 / 4 = 625 (this is sufficient)
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Re: Variable in Exponent DS problem [#permalink]

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23 Nov 2011, 10:51
St1: $$5^k >600$$ ----insufficient as we don't know if $$5^k <1000$$

St2: $$5 ^ {k-1} = 5^K - 500$$
$$500 = 5^k (1- \frac{1}{5})$$
$$500 (\frac{5}{4})= 5^k$$ ----sufficient as we can determine if $$5^k <1000$$

hence B!
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Re: Variable in Exponent DS problem [#permalink]

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22 May 2014, 22:11
VeritasPrepKarishma wrote:
tonebeeze wrote:
I would appreciate it if someone could walk me through this problem. Thanks

Is $$5^k$$ less than 1000?

1. $$5^k+1 > 3000$$
2. $$5^k-1 = 5^k -500$$

I am assuming the question is:

Is $$5^k$$ less than 1000?

1. $$5^{k+1} > 3000$$
2. $$5^{k-1} = 5^k -500$$

$$5^4 = 625$$ and $$5^5 = 3125$$ (even if you do not know this, it is fine. You don't need to calculate. Just observe that 625*5 will be greater than 3000)

Statement 1: $$5^{k+1} > 3000$$
This means k + 1 is greater than 4 so k is greater than 3 (It doesnt mean that k + 1 is at least 5 because the question doesn't say that k is an integer. k + 1 could be 4.999 making k = 3.999) Since k can take values less than 4 and more than 4, 5^k could be less than 1000 or more than 1000. Not sufficient.

Statement 2: $$5^{k-1} = 5^k -500$$
Re-arrange: $$500 = 5^k -5^{k-1}$$
$$5^3 *4 = 5^{k-1}(5 - 1)$$
Hence k - 1 = 3 and k = 4
So $$5^k$$ = 625 which is less than 1000. Answer is 'Yes'. Sufficient.

I can't get why are we talking about decimals in the first statement. I don't know if I'm doing something wrong here...

I did it like this
$$5^{k+1} > 3000$$
$$k+1 \geq 5$$
$$k \geq 4$$

$$k= 4$$ then $$5^4 = 625$$
$$k=5$$ then $$5^5 = 3000$$ (approx)
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Re: Variable in Exponent DS problem [#permalink]

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23 May 2014, 01:12
Expert's post
1
This post was
BOOKMARKED
b2bt wrote:
VeritasPrepKarishma wrote:
tonebeeze wrote:
I would appreciate it if someone could walk me through this problem. Thanks

Is $$5^k$$ less than 1000?

1. $$5^k+1 > 3000$$
2. $$5^k-1 = 5^k -500$$

I am assuming the question is:

Is $$5^k$$ less than 1000?

1. $$5^{k+1} > 3000$$
2. $$5^{k-1} = 5^k -500$$

$$5^4 = 625$$ and $$5^5 = 3125$$ (even if you do not know this, it is fine. You don't need to calculate. Just observe that 625*5 will be greater than 3000)

Statement 1: $$5^{k+1} > 3000$$
This means k + 1 is greater than 4 so k is greater than 3 (It doesnt mean that k + 1 is at least 5 because the question doesn't say that k is an integer. k + 1 could be 4.999 making k = 3.999) Since k can take values less than 4 and more than 4, 5^k could be less than 1000 or more than 1000. Not sufficient.

Statement 2: $$5^{k-1} = 5^k -500$$
Re-arrange: $$500 = 5^k -5^{k-1}$$
$$5^3 *4 = 5^{k-1}(5 - 1)$$
Hence k - 1 = 3 and k = 4
So $$5^k$$ = 625 which is less than 1000. Answer is 'Yes'. Sufficient.

I can't get why are we talking about decimals in the first statement. I don't know if I'm doing something wrong here...

I did it like this
$$5^{k+1} > 3000$$
$$k+1 \geq 5$$
$$k \geq 4$$

$$k= 4$$ then $$5^4 = 625$$
$$k=5$$ then $$5^5 = 3000$$ (approx)

$$5^{k+1} > 3000$$ --> $$5*5^{k} > 3000$$ --> $$5^{k} > 600$$. Now, we are NOT told that k is an integer, thus we cannot say that $$k\geq{4}$$. For example, $$5^{3.99}\approx{615}$$, thus k could be 3.99.

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.

OPEN DISCUSSION OF THIS QUESTION IS HERE: is-5-k-less-than-1-000-1-5-k-128055.html
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Re: Variable in Exponent DS problem   [#permalink] 23 May 2014, 01:12
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