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If n is an integer and 5^n > 4,000,000, what is the least

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If n is an integer and 5^n > 4,000,000, what is the least  [#permalink]

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New post 19 Feb 2012, 19:02
2
17
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A
B
C
D
E

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If n is an integer and 5^n > 4,000,000, what is the least possible value of n?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

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Re: If n is an integer and 5^n > 4,000,000, what is the least  [#permalink]

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New post 21 Feb 2012, 14:05
7
3
Answer should be D. Here is how I solved it.

Original Statement: \(5^n>4,000,000\)
so \(5^n>4*10^6\)
so \(5^n>4*2^6*5^6\)

dividing by 5^6 on both sides leads to:

so \(5^{n-6}>2^8\)
so \(5^{n-6}>256\)

Now we know that \(5^3=125\)

so it has to be that the number is \(5^4\), the least number that is greater than \(256\)
so \(n-6=4\)
so \(n=10\)

Hence D.
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Re: If n is an integer and...  [#permalink]

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New post 19 Feb 2012, 21:42
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vix wrote:
If n is an integer and 5^n > 4,000,000, what is the least possible value of n?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11


Make prime factorization of \(4,000,000=2^2*1,000,000=2^2*(10^6)=2^8*5^6=256*5^6\).

So, we have: \(5^n>256*5^6\) --> \(5^4=625>256\) (5^3=125<256), so \(5^n\) must be at least equal to \(5^4*5^6=5^{10}\) to be more than \(256*5^6\).

Hence, the least possible value of n is 10.

Answer: D.

Hope it's clear.
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Re: If n is an integer and...  [#permalink]

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New post 09 Aug 2012, 13:17
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I may be wrong here but with the question written as 5^n <4000000,
and wanting the least possible value for n, it would just be the lowest answer choice (7) since it will easily be under 4000000
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Re: If n is an integer and...  [#permalink]

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New post 09 Aug 2012, 13:40
sdrobbi wrote:
I may be wrong here but with the question written as 5^n <4000000,
and wanting the least possible value for n, it would just be the lowest answer choice (7) since it will easily be under 4000000



Really sorry, I made a mistake on the sign. Corrected, thx..
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Re: If n is an integer and...  [#permalink]

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New post 09 Aug 2012, 14:59
arthuro69 wrote:
If n is an integer and 5^n > 4 000 000 what is the least possible value of n ?

A-7
B-8
C-9
D-10
E-11


How would you guys answer quickly this question ?


Merging similar topics. Please refer to the solutions above.

P.S. Please read and follow: rules-for-posting-please-read-this-before-posting-133935.html (rule #7).
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If n is an integer and 5^n > 4,000,000, what is the least possib  [#permalink]

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New post 29 Sep 2012, 23:18
Hi everyone,
This is an old thread, however I do have a question. Are we supposed to know value such as 2^9 off the top of our head?
Thanks for your help
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Re: If n is an integer and 5^n > 4,000,000, what is the least po  [#permalink]

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New post 01 Oct 2012, 06:42
mockney wrote:
Hi everyone,
This is an old thread, however I do have a question. Are we supposed to know value such as 2^9 off the top of our head?
Thanks for your help


Yes, it's good to know the powers of 2 till 2^10=1024. Check here for a solution: if-n-is-an-integer-and-5-n-4-000-000-what-is-the-least-127815.html#p1046805

Hope it helps.
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Re: If n is an integer and 5^n > 4,000,000, what is the least  [#permalink]

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New post 20 Nov 2013, 12:42
here is my approach -
5^n > 4000000.
=> (10/2)^n > 4000000 .. now 10^10 will have 1 followed by 10 zeros. Dividing by 2^10 = 1024, the expression will become 10^7/1.024.
its simple to see 10^9/512 means 1000000/5.12 < 4000000.

If you know the values of power of 2 (which I think is really helpful), you might do calculation this way in mind implicitly.
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If n is an integer and 5^n > 4,000,000, what is the least  [#permalink]

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New post 03 Mar 2016, 22:03
My approach

5^2 = 25
5^3 = 125
5^4 = 625
5^5 = 3.X thousand
5^6 = 16-18K
5^7 = ~80k
5^8 = ~450K
5^9 = ~2 mil
5^10 = less than 40 mil, therefore the highest answer is 11. E
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Re: If n is an integer and 5^n > 4,000,000, what is the least  [#permalink]

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New post 09 Mar 2016, 23:41
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Re: If n is an integer and 5^n > 4,000,000, what is the least  [#permalink]

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New post 02 Jul 2017, 10:17
5^n > 4000000.
=> (10/2)^n > 4000000 .. now 10^10 will have 1 followed by 10 zeros. Dividing by 2^10 = 1024, the expression will become 10^7/1.024.
its simple to see 10^9/512 means 1000000/5.12 < 4000000.

SO , ans is 10. D
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Re: If n is an integer and 5^n > 4,000,000, what is the least  [#permalink]

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New post 12 Jul 2017, 09:46
vix wrote:
If n is an integer and 5^n > 4,000,000, what is the least possible value of n?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11



I thing we can convert the above question as

\(5^n > 4,000,000 ..... sqrt(5^n) > 2,000 .....5^n/2 > 2,000\)

now it is much easier to find the value of n/2 so it \(5^n/2\) greater than 2000.

\(5^n/2 > 2,000 ... 5^5 > 2000\) so n/2 = 5 and n = 10.
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Re: If n is an integer and 5^n > 4,000,000, what is the least  [#permalink]

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New post 16 Jul 2017, 17:24
vix wrote:
If n is an integer and 5^n > 4,000,000, what is the least possible value of n?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11


Let’s break 4,000,000 into primes:

4,000,000 = 4 x 10^6 = 4 x 2^6 x 5^6 = 256 x 5^6.

So, we have:

5^n > 256 x 5^6

We see that n will definitely have to be greater than 6. We have to figure out what power of 5 is going to get us to a number greater than 256. We see that 5^3 = 125 isn’t big enough, but 5^4 = 625 fulfills our requirement of being greater than 256. Thus, n will be (6 + 4) = 10, because it is true that 5^10 > 256 x 5^6.

Answer: D
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Re: If n is an integer and 5^n > 4,000,000, what is the least  [#permalink]

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Re: If n is an integer and 5^n > 4,000,000, what is the least   [#permalink] 02 Jan 2019, 00:05
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