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OG -106

This topic has 3 expert replies and 2 member replies
neeg Junior | Next Rank: 30 Posts Default Avatar
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OG -106

Post Sat Mar 30, 2013 11:43 am
What is the smallest integer n for which 25^n > 5^12?

a)6
b)7
c)8
d)9
e)10

If I try solving this by rewriting the right side rather than the left side as a common base i.e. 5 ^12 --> 25^ 11 then n would have to be greater than 11 ?

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GMAT/MBA Expert

Post Sun Mar 31, 2013 7:34 pm
neeg wrote:
What is the smallest integer n for which 25^n > 5^12?

a)6
b)7
c)8
d)9
e)10

If I try solving this by rewriting the right side rather than the left side as a common base i.e. 5 ^12 --> 25^ 11 then n would have to be greater than 11 ?
An alternate approach would be to plug in the answers, starting with the smallest answer choice.

If n=6, we get:
25⁶ > 5¹²
(5²)⁶ > 5¹²
5¹² > 5¹².
Since n=6 makes the lefthand side EQUAL to the righthand side, the smallest possible value of n must be 7.

The correct answer is B.

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neeg Junior | Next Rank: 30 Posts Default Avatar
Joined
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Posted:
12 messages
Post Sun Mar 31, 2013 5:27 pm
Brilliant as always. I can see the error, Thank you.

Anju@Gurome wrote:
neeg wrote:
If I try solving this by rewriting the right side rather than the left side as a common base i.e. 5 ^12 --> 25^ 11 then n would have to be greater than 11 ?
Your mistake is there.
Arithmetic of powers do not work that way.
5^12 = 5*(5^11) NOT 25^11
Or, 5^12 = (5^6)*(5^6) = (5*5)^6 = 25^6

To clear your confusion, let us go back to the very definition of powers,
a^n = a multiplied with it self n times = a*a*a*... (n times)
a^m = a multiplied with it self m times = a*a*a*... (m times)

Now,
    a^(m + n) = (a^m)*(a^n)
    a^(mn) = (a^m)^n = (a^n)^m

Hope that helps.

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GMAT/MBA Expert

Post Sat Mar 30, 2013 12:04 pm
neeg wrote:
What is the smallest integer n for which 25^n > 5^12?

a)6
b)7
c)8
d)9
e)10

First recognize that 5^12 = (5^2)^6 = 25^6
What is the smallest integer n for which 25^n > 25^6?
We need n>6
So, n = 7 is the smallest integer.
Answer = B

Cheers,
Brent

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rintoo22 Senior | Next Rank: 100 Posts Default Avatar
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Target GMAT Score:
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Post Sat Mar 30, 2013 12:05 pm
25^n > 5^12 => 5^2n > 5^12

Therefore 2n=12 => n=6. So for n=6 both sides will match.

But the question asks the smallest integer n for which 25^n > 5^12. So the the left hand side shodl be greater. So the answer in my opinion should be b.7.

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Anju@Gurome GMAT Instructor
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Post Sat Mar 30, 2013 10:33 pm
neeg wrote:
If I try solving this by rewriting the right side rather than the left side as a common base i.e. 5 ^12 --> 25^ 11 then n would have to be greater than 11 ?
Your mistake is there.
Arithmetic of powers do not work that way.
5^12 = 5*(5^11) NOT 25^11
Or, 5^12 = (5^6)*(5^6) = (5*5)^6 = 25^6

To clear your confusion, let us go back to the very definition of powers,
a^n = a multiplied with it self n times = a*a*a*... (n times)
a^m = a multiplied with it self m times = a*a*a*... (m times)

Now,
    a^(m + n) = (a^m)*(a^n)
    a^(mn) = (a^m)^n = (a^n)^m

Hope that helps.

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Quant Expert, Gurome

Backup Methods : General guide on plugging, estimation etc.
Wavy Curve Method : Solving complex inequalities in a matter of seconds.

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