If k is an integer and 5^k

Question

If k is an integer and  5^k < 20,000  what is the greatest possible value of k ?

(A) 6 
(B) 7 
(C) 8 
(D) 9 
(E) 10

GMATinsight · Accepted Answer

5^k < 20,000

Let's Prime factorize the bigger number

5^k < 2*10^4

5^k < 2^5*5^4

dividing both sides by  5^4

5^{k-4} < 2^5

5^{k-4} < 32

Now, we know that  5^1 = 5  and  5^2 = 25  and  5^3 = 125

i.e.  (k-4)_{max} = 2

i.e.  {k}_{max} = 6

Answer: Option A

Kinshook · Answer

Asked: If k is an integer and  5^k < 20,000  what is the greatest possible value of k ?

5^6 = 625*25 = 15,625 <20,000
5^7 = 15500*5 = 78,125 >20,000

k_{max} = 6

IMO A

Anvesh99 · Answer

Hello  GMATinsight

Could you explain once again from below step onwards. I did not get, how you got K=6

"  Now, we know that  5^1 = 5  and  5^2 = 25  and  5^3 = 125

i.e.  (k-4)_{max} = 2

i.e.  {k}_{max} = 6  "

GMATinsight · Answer

I hope you are clear till the following step

5^{k-4} < 32

now, we already know that  5^1 = 5  and  5^2 = 25  and  5^3 = 125

i.e.  5^3  is NOT less than 32 so k-4 can NOT be 3 instead k-4 has to be less than 3

Also we see that  5^3  is less than 32 i.e. k-4 can be equal to 2

hence maximum value of k-4 can be 2

k-4 = 2
then k = 2+4 = 6

i.e. if maximum of k-4 is 2 then k in that case k will be 2+4

hence k max can be 6

ThatDudeKnows · Answer

Anvesh99

The approach to this question may depend on your fluency with exponents. If you're comfortable going about it as  GMATinsight  shows, that's awesome, as it certainly works. But I think it's also worthwhile to get good at thinking through questions in a way that may help you avoid a lot of the math altogether. I think you can probably do everything that follows in your head in about 10 seconds with minimal risk of making a silly mistake, and you'll be really sure you have the right answer. I'd just skip the "real" math here.

5^3 = 125
So, 5^6 = 125*125
100 * 125 is 12,500. and 25 * 125 is one-fourth of that, which we can just ballpark to 3,000. That makes 125 * 125 = roughly 15,500. 
Are we going to be able to multiply that by 5 again and still be under 20,000? No way.

Answer choice A.

bumpbot · Answer

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If k is an integer and 5^k < 20,000 what is the greatest possible valu

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