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Please Help solve this! QID 150001 Princeton

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maggiejdyer Just gettin' started! Default Avatar
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Please Help solve this! QID 150001 Princeton Post Sun Jan 04, 2015 8:34 am
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  • Lap #[LAPCOUNT] ([LAPTIME])
    Need help solving!

    Hi, this is question ID 150001 from the free Princeton Review Practice test. There is no explanation given to how to solve the problem, other than "this is the answer when plugging in." Can someone walk through the steps here?

    Thanks!

    If x, y, and z are positive integers, x is a factor of 2y, and 3y is a factor of z, which of the following must also be an integer?


    y/x


    2y/6


    xy/3z


    zx/3y


    zy/3x

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    Post Sun Jan 04, 2015 9:51 am
    maggiejdyer wrote:
    If x, y, and z are positive integers, x is a factor of 2y, and 3y is a factor of z, which of the following must also be an integer?
    A) y/x
    B) 2y/6
    C) xy/3z
    D) zx/3y
    E) zy/3x
    We want to find which answer choice MUST be an integer. So, let's ELIMINATE the answer choices that NEED NOT be an integer.

    First, we'll find some values for x, y and z that satisfy the given conditions.

    x is a factor of 2y
    How about x = 2 and y = 3

    3y is a factor of z
    Since y = 3, we need (3)(3) to be a factor of z.
    How about z = 9

    Now, we'll plug x = 2, y = 3 and z = 9 into each answer choice to see what we get:

    A) y/x = 3/2. This is NOT an integer. So, we can ELIMINATE A
    B) 2y/6 = 6/6. This IS an integer. KEEP B
    C) xy/3z = 6/27. This is NOT an integer. So, we can ELIMINATE C
    D) zx/3y = 18/9 = 2. This IS an integer. KEEP D
    E) zy/3x = 27/6. This is NOT an integer. So, we can ELIMINATE E

    So, the correct answer is EITHER B OR D.

    To determine which of these is correct, we must test a DIFFERENT set of values.
    Another set of values that satisfy the given conditions is: x = 2, y = 2, and z = 6
    Now, we'll plug x = 2, y = 2, and z = 6 into the two remaining answer choices to see what we get:

    B) 2y/6 = 4/6. This is NOT an integer. ELIMINATE B

    At this point, the answer MUST be D. We need not waste any time plugging values into answer choice D

    Cheers,
    Brent

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    Post Sun Jan 04, 2015 10:29 am
    Hi maggiejdyer,

    I'm a big fan of TESTing VALUES (the approach that Brent used) - it's perfect for these types of questions.

    The answer choices are written in such a way though that you can answer this question using Number Property rules....

    In the prompt, we're told that "3Y is a factor of Z." This is just a wordy way of saying Z/3Y is an INTEGER. We're also told that X, Y and Z are positive integers.

    Looking at the answers, what do you notice about answer D?

    ZX/3Y....

    It's interesting that the letters in in the numerator are NOT in alphabetical order. I wonder why THAT is? Is it to hide the fact that this answer has Z/3Y in it? We already know that Z/3Y is an integer and we know that X is an integer too.

    Answer D literally translates into (integer)(integer)....which is ALWAYS an integer.

    Final Answer: D

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    maggiejdyer Just gettin' started! Default Avatar
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    Post Sun Jan 04, 2015 11:45 am
    Thank you both! So quick and thorough, appreciate it!

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    Post Mon Jan 05, 2015 6:42 am
    Quote:
    If x, y, and z are positive integers, x is a factor of 2y, and 3y is a factor of z, which of the following must also be an integer?
    A) y/x
    B) 2y/6
    C) xy/3z
    D) zx/3y
    E) zy/3x
    One can also use the approach to refute the options and end up getting the correct answer
    x is a factor of 2y, and 3y is a factor of z, which of the

    A) y/x, @y=3 and x=2, x is factor of 2y but y/x is not an integer so ELIMINATED
    B) 2y/6 i.e. y/3, @y=5 and x=5, x is factor of 2y but y/3 is not an integer so ELIMINATED
    C) xy/3z, @y=3, x=2 and z=45, all conditions are satisfied but xy/3z is not an integer so ELIMINATED
    D) zx/3y
    E) zy/3x, @y=3, x=2 and z=9, all conditions are satisfied but zy/3x is not an integer so ELIMINATED

    Answer: Option D

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    Post Mon Jan 05, 2015 10:38 am
    If we want to do something OTHER than plug in, we could notice the following.

    Since 3y is a factor of z, we know that z/3y = an integer.

    Since answer (D) is just x * (z/3y), we know that reduces to x * (an integer), which must be an integer! Success!

    One of the nice things about working through these problems conceptually is that they become very quick once you get the hang of the concepts, whereas plugging in always takes a long time.

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    Post Sat Jun 18, 2016 4:14 am
    I solved it by plugging in the values

    x is factor of 2y
    ie 2y/x=integer lets take x=2 and y=5-why 5 because if we multiply by 2 we get 10 which is divisible by 2.

    3y is a factor of z. ie z/3y=integer.

    we have already taken y=3 therefore 3y=15. Lets chose a value of z which is divisible by 15 ie let us take 30.

    so x=2, y=5 and z=30.

    plug in the values in answer choice to get D as the answer.

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