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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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Posted:
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OG -106

Post Sat Mar 30, 2013 11:43 am
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    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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    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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    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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    neeg Junior | Next Rank: 30 Posts Default Avatar
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    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.

    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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