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

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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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8
6
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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Posts: 58446
Re: If n is an integer and...  [#permalink]

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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.

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]

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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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Joined: 02 Sep 2009
Posts: 58446
Re: If n is an integer and 5^n > 4,000,000, what is the least po  [#permalink]

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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]

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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.

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

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_________________ Re: If n is an integer, and 5^n > 4,000,000. What is the least possible   [#permalink] 30 Sep 2019, 23:41
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