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a% of b% of c an integer?

This topic has 5 expert replies and 3 member replies
j_shreyans Legendary Member Default Avatar
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a% of b% of c an integer?

Post Sun Jun 07, 2015 5:22 am
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    If a, b, and c are positive integers such that a < b < c, is a% of b% of c an integer?

    (1) b=(a/100)^-1

    (2) c = 100b

    Statement 1 = b=(a/100)^-1

    b=1/(a/100) so b = 100/a

    if i put b=100/a in our target question then it will give a/100 X 100/a X c

    so we will have only C and a,b,and c is positive integers it's given .

    so this statement should be true.

    Please advise and correct me.

    Thanks,

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    theCEO Master | Next Rank: 500 Posts Default Avatar
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    Post Sun Jun 07, 2015 5:59 am
    j_shreyans wrote:
    If a, b, and c are positive integers such that a < b < c, is a% of b% of c an integer?

    (1) b=(a/100)^-1

    (2) c = 100b

    Statement 1 = b=(a/100)^-1

    b=1/(a/100) so b = 100/a

    if i put b=100/a in our target question then it will give a/100 X 100/a X c

    so we will have only C and a,b,and c is positive integers it's given .

    so this statement should be true.

    Please advise and correct me.

    Thanks,
    b% of c = (bc)/100
    a% of b% of c = (abc)/10,000
    Is (abc)/10,000 an integer?

    (1) b=(a/100)^-1
    1 equation and 2 unknowns, cant solve so statement is insufficent

    (2) c = 100b
    1 equation and 2 unknowns, cant solve so statement is insufficent

    Combining both
    b=(a/100)^-1 = 100/a
    ab = 100
    c = 100b

    (abc)/10,000 = 10,000b / 10,000 = b which is an integer
    statement is sufficent
    answer = c

    Post Sun Jun 07, 2015 6:24 am
    The posted problem has a typo.
    It should read as follows:

    Quote:
    If a, b, and c are positive integers such that a < b < c, is a% of b% of c an integer?

    (1) b = (a/100)^(-1)

    (2) c = 100^b
    Does (a/100) * (b/100) * c = integer?

    Test integer values such that a < b < c.

    Statement 1: b = (a/100)¯¹
    Thus, b = 100/a.

    Test the smallest possible value for a.
    Case 1: a=1
    Here, b =100/1 = 100.
    In this case, (a/100) * (b/100) * c = 1/100 * 100/100 * c = c/100.

    If c = 200, then c/100 = 2, which is an integer.
    If c = 201, then c/100 = 201/100, which is not an integer.
    INSUFFICIENT.

    Statement 2: c = 100^b
    Test the smallest possible value for b.
    Case 2: b=2
    Here, a=1 and c = 100² = 10000.
    In this case, (a/100) * (b/100) * c = 1/100 * 2/100 * 10000 = 2, which is an integer.

    Test an extreme value for b.
    Case 3: b=100
    Here, c = 100¹⁰⁰.
    In this case, (a/100) * (b/100) * c = a/100 * 10/100 * 100¹⁰⁰ = 10a * 100⁹⁸, which is an integer.

    Cases 2 and 3 illustrate that -- given that c = 100^b -- (a/100) * (b/100) * c will always be equal to an integer value.
    SUFFICIENT.

    The correct answer is B.

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    Last edited by GMATGuruNY on Fri Jun 12, 2015 9:55 pm; edited 1 time in total

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    Post Sun Jun 07, 2015 6:47 am
    Quote:
    If a, b, and c are positive integers such that a < b < c, is a% of b% of c an integer?

    (1)b = (a/100)^-1
    (2) c = 100^b
    Target question: Is a% of b% of c an integer?

    This is a great candidate for rephrasing the target question.
    Aside: We have a free video with tips on rephrasing the target question: http://www.gmatprepnow.com/module/gmat-data-sufficiency?id=1100

    a% of b% of c is the same as (a/100)(b/100)(c), which equals abc/10,000
    So, we can rephrase the target question as follows:
    REPHRASED target question: Is abc/10,000 an integer?

    We can REPHRASE the target question even further...
    RE-REPHRASED target question: Is abc a multiple of 10,000?

    Statement 1: b = (a/100)^-1
    In other words, b = 100/a
    There are several values of a, b and c that satisfy this condition. Here are two:
    Case a: a = 1, b = 100 and c = 1000, in which case abc = 100,000. Here, abc IS a multiple of 10,000
    Case b: a = 1, b = 100 and c = 101, in which case abc = 10,100. Here, abc is NOT a multiple of 10,000
    Since we cannot answer the RE-REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

    Statement 2: c = 100^b
    IMPORTANT: We are told that a, b and c are POSITIVE INTEGERS and that a < b < c
    So, we can be certain that b > 2.
    If b is greater than or equal to 2, then c (which equals 100^b) can equal 10,000 or 1,000,000 or 100,000,000 and so on.
    Notice that ALL of these possible values of c are multiples of 10,000
    So, if c is a multiple of 10,000, then abc MUST be a multiple of 10,000
    Since we can answer the RE-REPHRASED target question with certainty, statement 2 is SUFFICIENT

    Answer = B

    Cheers,
    Brent

    For even more information on rephrasing the target question, you can read this article I wrote for BTG: http://www.beatthegmat.com/mba/2014/06/30/rephrasing-the-target-question

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    Post Sun Jun 07, 2015 6:49 am
    Quote:
    if i put b=100/a in our target question then it will give a/100 X 100/a X c
    so we will have only C and a,b,and c is positive integers it's given .
    Be careful, j_shreyans - you're right that plugging the first statement into our question will cancel a and b. But as theCEO pointed out, our target question is really "is (abc)/10,000 an integer?"

    You seemed to have interpreted the question as "a% of b of c" rather than "a% of b% of c." If we substitute (a/100)^-1 for b, we would get (a/100)((100/a)/100)(c). When we simplify, we're still left with c/100. Since the only constraint was a < b < c, we can't know if c is divisible by 100. Insufficient.

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    Post Sun Jun 07, 2015 6:54 am
    theCEO wrote:
    (1) b=(a/100)^-1
    1 equation and 2 unknowns, cant solve so statement is insufficent

    (2) c = 100b
    1 equation and 2 unknowns, cant solve so statement is insufficent
    Be careful, theCEO! The GMAT will often mess with our expectations that we need 2 equations for 2 unknowns, 3 equations for 3 unknowns, etc. Remember that CONSTRAINTS are always given for a reason, and you didn't use the constraint that a, b, & c are positive and a < b < c. Since this is the case, as Mitch pointed out, the minimum value of b is 2, and thus the minimum value of 100^b = 10,000. This is sufficient.

    So remember to always ask yourself - why did they include this restraint?

    For more on how the GMAT breaks the "2 unknowns, 2 equations" rule, see here: https://www.manhattanprep.com/gmat/blog/2015/02/19/high-school-algebra-wrong-gmat-breaks-systems-equations-rules/

    _________________


    Ceilidh Erickson
    Manhattan Prep GMAT & GRE instructor
    EdM in Mind, Brain, and Education
    Harvard Graduate School of Education


    Manhattan Prep instructors all have 99th+ percentile scores and expert teaching experience.
    Sign up for a FREE TRIAL, and learn why we have the highest ratings in the GMAT industry!

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    theCEO Master | Next Rank: 500 Posts Default Avatar
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    Post Sun Jun 07, 2015 7:10 am
    ceilidh.erickson wrote:
    theCEO wrote:
    (1) b=(a/100)^-1
    1 equation and 2 unknowns, cant solve so statement is insufficent

    (2) c = 100b
    1 equation and 2 unknowns, cant solve so statement is insufficent
    Be careful, theCEO! The GMAT will often mess with our expectations that we need 2 equations for 2 unknowns, 3 equations for 3 unknowns, etc. Remember that CONSTRAINTS are always given for a reason, and you didn't use the constraint that a, b, & c are positive and a < b < c. Since this is the case, as Mitch pointed out, the minimum value of b is 2, and thus the minimum value of 100^b = 10,000. This is sufficient.

    So remember to always ask yourself - why did they include this restraint?

    For more on how the GMAT breaks the "2 unknowns, 2 equations" rule, see here: https://www.manhattanprep.com/gmat/blog/2015/02/19/high-school-algebra-wrong-gmat-breaks-systems-equations-rules/
    Thanks alot Ceilid!

    theCEO Master | Next Rank: 500 Posts Default Avatar
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    Post Fri Jun 12, 2015 7:00 pm
    After reviewing this post, I realize that the question asked and the question Mitch solved are different.

    j_shreyans wrote:
    If a, b, and c are positive integers such that a < b < c, is a% of b% of c an integer?

    (2) c = 100b
    GMATGuruNY wrote:
    If a, b, and c are positive integers such that a < b < c, is a% of b% of c an integer?

    (2) c = 100^b

    Post Fri Jun 12, 2015 9:53 pm
    Good catch.
    This problem is from MGMAT.

    j_shreyans wrote:
    (2) c = 100b
    Here, statement 2 has a typo.
    In the original MGMAT problem, statement 2 reads as follows:

    GMATGuruNY wrote:
    (2) c = 100^b
    I've amended my post above to call attention to this difference.
    Even so:

    theCEO wrote:
    (2) c = 100b
    1 equation and 2 unknowns, cant solve so statement is insufficent
    Here, the portion in red mischaracterizes the task at hand.
    To answer the question stem -- Is a% of b% of c an integer? -- we do no have to solve.
    Rather, we have to determine whether abc is a multiple of 10,000.
    A clearer line of reasoning would be as follows:
    If a=1, b=2 and c=200, then abc is not a multiple of 10,000.
    If a=1, b=100, and c=10,000, then abc is a multiple of 10,000.

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    For more information, please email me at GMATGuruNY@gmail.com.

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