If n is an integer, and 5^n > 4,000,000. What is the least possible : GMAT Problem Solving (PS)
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# If n is an integer, and 5^n > 4,000,000. What is the least possible

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

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11 Jan 2010, 04:24
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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
[Reveal] Spoiler: OA
Joined: 20 Aug 2009
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Kudos [?]: 140 [4] , given: 69

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

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11 Jan 2010, 05:22
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$$4,000,000 = 2^2*1,000,000 = 2^2*10^6=2^2*2^6*5^6 = 2^8*5^6$$

$$5^n>2^8*5^6$$

$$5^{n-6}>256$$

$$625=5^4>256$$

$$n-6=4$$

$$n=10$$
Manager
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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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11 Jan 2010, 06:28
You can solve it with the same way you find LCM.

5 | 4,000,000
5 | 800,000
5 | 160,000
5 | 32,000
5 | 6,400
5 | 1,280
5| 256
5 | 56
5 | 11
2

We have 9 5's and 1x2. For $$5^n$$ to be bigger than 4,000,000, we therefore need n=10.
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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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14 Feb 2010, 09:51
shalva wrote:
$$4,000,000 = 2^2*1,000,000 = 2^2*10^6=2^2*2^6*5^6 = 2^8*5^6$$

$$5^n>2^8*5^6$$

$$5^{n-6}>256$$

$$625=5^4>256$$

$$n-6=4$$

$$n=10$$

nice approach. What is OA?
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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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14 Feb 2010, 11:15
IMO D -10

$$5^n > 5^6 * 4 * 2^6$$
$$5^n > 5^6 * 2^8$$
$$5^n > 5^6 * 256$$
$$5^n > 5^6 * 5^3 * \frac{256}{125}$$
$$5^n > 5^9* \frac{256}{125}$$ since $$\frac{256}{125}< 5$$

n = 10
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Re: If n is an integer, and 5^n > 4,000,000. What is the least possible   [#permalink] 14 Feb 2010, 11:15
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