Saturday, June 7, 2014

147: Exponents

If a, b are positive integers greater than 1 and ba=212 then what is the largest possible value of b?

from D. Marain

3 comments:

  1. I want to say 8 but that seems too obvious to me. Have I missed something?

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  3. I believe the only allowable possibilities are

    (2^2)^6 = (2^3)^4 = (2^4)^3 = (2^6)^2

    Since the only prime factor of 2^12 is 2, the same must be true for b, and the only ways to write 12 as an ordered product of integers greater than 1 is

    (2)(6) = (3)(4) = (4)(3) = (6)(2)

    In other words, the possibilities are:

    1. a = 2^2 = 4 and b = 6

    2. a = 2^3 = 8 and b = 4

    3. a = 2^4 = 16 and b = 3

    4. a = 2^6 = 64 and b = 2

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