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# Is 5^k less than 1,000? (1) 5^(k+1) > 3,000

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Intern
Joined: 20 Feb 2012
Posts: 30
Is 5^k less than 1,000? (1) 5^(k+1) > 3,000  [#permalink]

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23 Feb 2012, 23:42
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8
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Difficulty:

35% (medium)

Question Stats:

75% (02:00) correct 25% (02:30) wrong based on 485 sessions

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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: https://gmatclub.com/forum/is-5-k-less- ... l#p1209238
Math Expert
Joined: 02 Sep 2009
Posts: 52152
Re: Is 5^k less than 1,000? (1) 5^(k+1) > 3,000  [#permalink]

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23 Feb 2012, 23:54
3
5
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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Math Expert
Joined: 02 Sep 2009
Posts: 52152
Re: Is 5^k less than 1,000? (1) 5^(k+1) > 3,000  [#permalink]

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25 Jun 2013, 03:46
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Exponents: math-number-theory-88376.html

All DS Exponents questions to practice: search.php?search_id=tag&tag_id=39
All PS Exponents questions to practice: search.php?search_id=tag&tag_id=60

Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html

_________________
Manager
Joined: 21 Oct 2013
Posts: 185
Location: Germany
GMAT 1: 660 Q45 V36
GPA: 3.51
Re: Is 5^k less than 1,000? (1) 5^(k+1) > 3,000  [#permalink]

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11 Dec 2013, 21:24
Bunuel wrote:
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.

Hey Bunuel,

could you explain why you can factor out 5^k-1 from 5^k? I don't understand why that is possible.
Math Expert
Joined: 02 Sep 2009
Posts: 52152
Re: Is 5^k less than 1,000? (1) 5^(k+1) > 3,000  [#permalink]

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12 Dec 2013, 02:20
unceldolan wrote:
Bunuel wrote:
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.

Hey Bunuel,

could you explain why you can factor out 5^k-1 from 5^k? I don't understand why that is possible.

Operations involving the same bases:
Keep the base, add or subtract the exponent (add for multiplication, subtract for division)
$$a^n*a^m=a^{n+m}$$

$$5^{k-1}(5-1)=5^{k-1}*5-5^{k-1}=5^k-5^{k-1}$$

Theory on Exponents: math-number-theory-88376.html

All DS Exponents questions to practice: search.php?search_id=tag&tag_id=39
All PS Exponents questions to practice: search.php?search_id=tag&tag_id=60

Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html

Hope this helps.
_________________
Manager
Joined: 21 Oct 2013
Posts: 185
Location: Germany
GMAT 1: 660 Q45 V36
GPA: 3.51
Re: Is 5^k less than 1,000? (1) 5^(k+1) > 3,000  [#permalink]

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12 Dec 2013, 03:25
Ok now I get it, thank you very much!
Intern
Joined: 23 Aug 2014
Posts: 34
GMAT Date: 11-29-2014
Re: Is 5^k less than 1,000? (1) 5^(k+1) > 3,000  [#permalink]

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12 Nov 2014, 00:37
If we know the first few powers of 5 it gets real easy.
for example $$5^2=25, 5^3=125, 5^4=25^2=625, 5^5=3125$$

I read somewhere that a gmat taker should ideally know these
- decimal value of common fractions- 1/2, 1/3, 1/4, 1/5- in turn we'll know 2/3, 2/5, 3/4, 1/8...
- factorials till 6! maybe
- perfect squares (say till 25)
- first 5 powers of 2,3,4,5

Math Expert
Joined: 02 Sep 2009
Posts: 52152
Re: Is 5^k less than 1,000? (1) 5^(k+1) > 3,000  [#permalink]

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13 Jan 2018, 11:28
BANON wrote:
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: https://gmatclub.com/forum/is-5-k-less- ... l#p1209238
_________________
Re: Is 5^k less than 1,000? (1) 5^(k+1) > 3,000 &nbs [#permalink] 13 Jan 2018, 11:28
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# Is 5^k less than 1,000? (1) 5^(k+1) > 3,000

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