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

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vix
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If n is an integer and 5^n > 4,000,000, what is the least [#permalink] New post  19 Feb 2012, 19:02
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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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omerrauf
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Re: If n is an integer and 5^n > 4,000,000, what is the least [#permalink] New post  21 Feb 2012, 14:05
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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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Bunuel
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Re: If n is an integer and... [#permalink] 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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If n is an integer and 5^n > 4,000,000, what is the least possib [#permalink] 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] New post  01 Oct 2012, 06:42
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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] New post  16 Jul 2017, 17:24
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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


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] New post  26 Nov 2022, 22:43
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


Alternative way:

5=10/2

For 7 zeroes we need 10^7 and need to adjust 2^7 in the denominator, thus we need to add 3 more in power in order to maintain its value of more than 4,000,000.
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