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Arithmetic - Properties of Numbers: If n = ...

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wied81 Junior | Next Rank: 30 Posts Default Avatar
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Arithmetic - Properties of Numbers: If n = ...

Post Sat Apr 28, 2012 5:37 pm
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    This is from OG 13 #117 in Problem Solving:

    If N = 3^8 - 2^8, which of the following is NOT a factor of n?

    A) 97

    B) 65

    C) 35

    D) 13

    E) 5

    OA: C


    The book posts a pretty obscure way to solve, would like to hear others opinions on this question.

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    Post Sat Apr 28, 2012 6:18 pm
    If N = 3^8 - 2^8, which of the following is NOT a factor of n?

    3^8 - 2^8 = (3^4-2^4)(3^4+2^4) - a^2 -b^2 = (a+b)*(a-b)
    3^8 - 2^8 = (3^2-2^2)(3^2+2^2)(3^4+2^4) - a^2 -b^2 = (a+b)*(a-b)
    3^8 - 2^8 = (3-2)(3+2)(3^2+2^2)(3^4+2^4)
    3^8 - 2^8 = (1)(5)(13)(97), so 97,65(13*5),13,5 are factors of 3^8 - 2^8. The only number left is 35, which isn't a factor of 3^8 - 2^8

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    Post Sun Apr 29, 2012 9:27 am
    3^8 - 2^8=(3^4-2^4)(3^4+2^4)=
    =(81-16)(81+16)=65*97 (a and b are out)

    65*97=5*13*97 (d and E are out)
    answ is C

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    Post Sat Jun 27, 2015 5:21 am
    wied81 wrote:
    This is from OG 13 #117 in Problem Solving:

    If N = 3^8 - 2^8, which of the following is NOT a factor of n?

    A) 97

    B) 65

    C) 35

    D) 13

    E) 5

    OA: C


    The book posts a pretty obscure way to solve, would like to hear others opinions on this question.
    Solution:

    It is very unlikely that a problem would require us to calculate 3^8 or 2^8, so we should approach this problem not as an arithmetic question but as an algebraic one.

    The first thing we should recognize is that we are being tested on the algebraic factoring technique called the "difference of squares." Recall that the general form of the difference of squares is:

    x^2 - y^2 = (x + y)(x - y)

    Similarly, we can treat 3^8 - 2^8 as a difference of squares, which can be expressed as:

    n = (3^4 + 2^4)(3^4 - 2^4)

    We can further factor 3^4 - 2^4 as an additional difference of squares, which can be expressed as:

    (3^2 + 2^2)(3^2 - 2^2)

    This finally gives us:

    n = 3^8 - 2^8 = (3^4 + 2^4)(3^2 + 2^2)(3^2 - 2^2)

    The numbers are now easy to calculate:

    n = (81 + 16)(9 + 4)(9 - 4)

    n = (97)(13)(5)

    We are being asked which of the answer choices is NOT a factor of n, which we have determined to be equal to the product (97)(13)(5). So we must find the answer choice that does not evenly divide into (97)(13)(5).

    We immediately see that 97, 13 and 5 are all factors of (97)(13)(5).

    This leaves us with 65 and 35. Notice that (97)(13)(5) = (97)(65). Thus, 65 also is a factor of n. Only 35 is not.

    Answer: C

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    Post Sat Jun 27, 2015 8:42 am
    I want to add something about the concept of algebraic manipulations without variables:

    The GMAT loves to test our algebra skills (like factoring), HOWEVER many students associate algebra with variables only, so they don't see that we can apply algebraic principles to numbers as well (which should make sense, since those variables are, indeed, representing numbers).

    So, for example, many students are fine with the following:
    - Factoring 6x + 3 to get 3(2x + 1)
    - Factoring 6x⁵ + 2x³ + 8x² to get 2x²(3x³ + x + 4)
    - Factoring x² + 5x + 6 to get (x + 2)(x + 3)
    - Factoring x² - y² to get (x + y)(x - y)

    On the GMAT, we need to recognize that we can also factor expressions that have no variables.
    So, for example, a GMAT question might ask us to evaluate 54² - 53²
    If we recognize that this is a difference of squares in the form x² - y², we can factor it to get:
    54² - 53² = (54 + 53)(54 - 53) = (107)(1) = 107

    Likewise, the expression 2¹⁰⁰ - 2⁹⁶ is no different from x¹⁰⁰ - x⁹⁶
    x¹⁰⁰ - x⁹⁶ = x⁹⁶(x⁴ - 1) in the exact same way that 2¹⁰⁰ - 2⁹⁶ = 2⁹⁶(2⁴ - 1)

    Cheers,
    Brent

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    Post Mon Jun 29, 2015 3:36 pm
    Here's an approach that doesn't involve difference of squares.

    If a number divides by 65, it must divide by 13 and by 5. So D and E *CANNOT* be the answers: if the number is not divisible by 5, it is also not divisible by 65. (Same logic for 13 and 65.) Now that we know the answer divides by 5 and 13, it must also divide by 65, so B CANNOT be the answer.

    Now we'll consider A and C. Since we know our number divides by 5, it will divide by 35 if and only if our number also divides by 7. (This is because 35 = 5 * 7).

    At this point, we recall that 3⁸ - 2⁸ will divide by 7 if 3⁸ and 2⁸ have the same remainder when divided by 7. 2⁸ = 256, which has a remainder of 4 when divided by 7. 3⁸ = 6561, which has a remainder of 2.

    So our number doesn't divide by 7, and hence can't divide by 35. We're done, and we don't need to bother testing A.

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    Post Thu Jul 02, 2015 4:24 am
    Factoring Formula - x² - y² to get (x + y)(x - y) is the key to solve this question.

    35 is the answer.

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