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"Length of integer" (prime number properties)

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JohnRapp Just gettin' started!
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"Length of integer" (prime number properties) Post Sun Nov 13, 2011 10:48 am
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    For any integer k > 1, the term “length of an integer” refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 X 2 X 2 X 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?

    A. 5
    B. 6
    C. 15
    D. 16
    E. 18

    I'm hoping this post can provide alternative explanations to the OA explanation.

    OA = (D) 16

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    rijul007 GMAT Destroyer!
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    Post Sun Nov 13, 2011 11:14 am
    for a number to have maximum length.. it should have the highest no of prime factors not necessarily distinct

    this number n should have max nuber of 2s as a prime factor

    x>1
    y>1

    x = 2^a [a is the length of x]
    y = 2^b [b is the length of y]

    x+3y<1000
    2^a + 3(2)^b <1000

    highest value of 2^a that can fit into this is 512

    2^a = 512
    a = 9

    3(2)^b <1000-512
    2^b < 488/3
    2^b < 162
    highest value of 2^b can be 128
    b = 7

    Maximum sum of length of a and length of b = a + b =9+7 = 16


    Option D

    rijul007 GMAT Destroyer!
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    Post Sun Nov 13, 2011 11:15 am
    for a number to have maximum length.. it should have the highest no of prime factors not necessarily distinct

    a number should have max nuber of 2s as a prime factor for the max length..

    x>1
    y>1

    x = 2^a [a is the length of x]
    y = 2^b [b is the length of y]

    x+3y<1000
    2^a + 3(2)^b <1000

    highest value of 2^a that can fit into this is 512

    2^a = 512
    a = 9

    3(2)^b <1000-512
    2^b < 488/3
    2^b < 162
    highest value of 2^b can be 128
    b = 7

    Maximum sum of length of x and length of y = a + b =9+7 = 16


    Option D

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