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Thread: Not An Integer

  1. #1
    Super Member harpazo's Avatar
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    Not An Integer

    If x , y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT necessarily an integer?

    A. (x + z)/z
    B. (y + z)/x
    C. (x + y)/z
    D. (xy/z)
    E. (yz/x)

    My answer is D.
    Book's answer is B.
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  2. #2
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    Re: Not An Integer

    Quote Originally Posted by harpazo View Post
    If x , y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT necessarily an integer?

    A. (x + z)/z
    B. (y + z)/x
    C. (x + y)/z
    D. (xy/z)
    E. (yz/x)

    My answer is D.
    Book's answer is B.
    How did you choose?

    "x is a factor of y" means what?
    "x is a multiple of z" means what?
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  3. #3
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    Re: Not An Integer

    Quote Originally Posted by harpazo View Post
    If x , y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT necessarily an integer?

    A. (x + z)/z
    B. (y + z)/x
    C. (x + y)/z
    D. (xy/z)
    E. (yz/x)

    My answer is D.
    Book's answer is B.
    Let $z=1,~x=2,~\&~y=4$ Is B. an integer? Is D.?
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  4. #4
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    Re: Not An Integer

    Saying that "x is a factor or y" means that there exist an integer, p, such that y= px. Saying that "x is a multiple of z" means that there exist an integer, q, such that x= qz. Write everything in terms of z:
    A) (x+ z)/z= (qz+ z)/z= (q+ 1)z/z= q+ 1, an integer.
    B) (y+ z)/x= (px+ z)/x= p+ z/x. Since x= qz, we can write that as p+ z/(qz)= p+ 1/q which is an integer if and only if q= 1.
    C) (x+ y)/z= (qz+ px)/z= (qz+ pqz)/z= (q+ pq)z/z= q+ pq, an integer.
    D) (xy)/z= ((qz)y)/z= (qy)z/z= qy, an integer.
    E) (yz)/x= ((px)z)/x= pz, an integer.
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  5. #5
    Super Member harpazo's Avatar
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    Re: Not An Integer

    Quote Originally Posted by TKHunny View Post
    How did you choose?

    "x is a factor of y" means what?

    This means y can be divided by x.

    "x is a multiple of z" means what?
    This means z can be divided by x.
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  6. #6
    Super Member harpazo's Avatar
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    Re: Not An Integer

    Quote Originally Posted by Plato View Post
    Let $z=1,~x=2,~\&~y=4$ Is B. an integer? Is D.?
    Let z = 1, x = 2, and y = 4.

    A. (x + z)/z

    (2 + 1)/1 = 3

    B. (y + z)/x

    (4 + 1)/2 = 5/2

    C. (x + y)/z

    (2 + 4)/1 = 6

    D. (xy/z)

    (2•4)/1 = 8

    E. (yz/x)

    (4•1)/2 = 2

    Are they not all integers?
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