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Thread: GMATPREP ? -Subtracting Exponents and Prime Factors - Does this shortcut reall work?

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    GMATPREP ? -Subtracting Exponents and Prime Factors - Does this shortcut reall work?

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    By the way, a very similar problem was on my REAL GMAT exam last month.

    The problem is as follows.
    What is the greatest prime factor of (4^17) - (2^28)?

    A) 2
    B) 3
    C) 5
    D) 7
    E) 11

    SPOILER: D


    My short cut is below in the spoiler. Do you think it works for all cases?
    SPOILER:
    Step 1:Create equal -> 4^17 - 2^28 is equivalent to 4^17 - 4^14
    Step 2: Subtract the exponents; 17-14 =3
    Step 3: Next, take an easier example to get to a 3 exponent difference still using the number 4 as your base.
    Example: 4^4 - 4^1 = 256 - 4 = 252
    Step 4: Now, working backwards from the answer choices, starting with (E), divide 252 by all answer choices and see what is the highest prime factor to divide into 252.
    (E) 252/11 = fraction -> Not a prime factor
    (D) 252/7 = 36 -> Is a Prime Factor, and highest (Correct Answer)
    (C) 252/5 = fraction -> Not a prime factor
    (B) 252/3 = fraction -> Not a prime factor
    (A) 252/2 = 126 -> Is a Prime Factor, but not the highest

    Oddly enough this works if you solve it for the base of 2 instead lets consider.

    Step 1:Create equal -> 4^17 - 2^28 is equivalent to
    (2^17)(2^17) - (2^28) which is also equal to (2^34) - (2^28)
    Step 2: Subtract the exponents; 34-28 = 6
    Step 3:
    Next, take an easier example to get to a 6 exponent difference still using the number 2 as your base.
    Example: 2^7 - 2^1 = 128 - 2 = 126
    Step 4: Now, working backwards from the answer choices, starting with (E), divide 126 by all answer choices and see what is the highest prime factor to divide into 126.
    (E) 126/11 = fraction -> Not a prime factor
    (D) 126/7 = 36 -> Is a Prime Factor, and highest (Correct Answer)
    (C) 126/5 = fraction -> Not a prime factor
    (B) 126/3 = 42 -> Is a Prime Factor, but not the highest
    (A) 126/2 = 126 -> Is a Prime Factor, but not the highest

    Seems to work with all examples. Am I crazy, or is this really a shortcut...inquiring minds (me) want to know!!!!!

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    Within my grasp! hitchhiker3000's Avatar
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    I don't know why you have to do all these to arrive at the answer.

    4^17 - 2^28 = 4^14(4^3 - 1)
    = 4^14(63)
    The biggest prime no. in the given list that will be a factor of this will be 7.
    as 63 = 7*9

    In a way what you have done is similar to this but involves more calculation.
    The difference between the exponents is 3 here also (4^3 - 4^0)

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    excellent approach hitchiker

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    What is the greatest prime factor of (4^17) - (2^28)?

    A) 2
    B) 3
    C) 5
    D) 7
    E) 11

    4^17-2^28 = 2^34-2^28 = 2^28 (2^6 - 1) = 2^28 * 63 = 2^28 *7*3*3

    Hence, 7

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