Knewton Brutal Math Challenge – Week 8

by on May 12th, 2010

The shirt is once again in play. Show your work.

If A,BCD is a four-digit positive integer such that A,BCD is equal to the product of the two-digit number AB and the three digit number ABC, which of the following two-digit numbers cannot be a factor of A,BCD? (Note: the first digit of a number cannot be equal to 0.)

i. AD
ii. CD
iii. CA

(A) None
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III

16 comments

  • Answer (D)

  • Take the number 1,020 which is equal to 102 x 10.

    A = 1
    B = 0
    C = 2
    D = 0

    i. AD = 10, which is a factor of 1,020
    ii. CD = 20, which is a factor of 1,020
    iii. CA = 21, which is not a factor of 1,020

    So the answer is (C).

  • Dave
    that ABCD : 1030

    so AD:10
    CD:30
    AC:13

    Only 10 is a factor of 1030.
    CD,AC are not factors of 1030

    pick D

  • ABC * AB = ABCD
    A B C
    A B
    -------
    A^2 (A^2+B^2) (B^2+AC) D

    A^2=A
    AB+AB=B
    B^2+AC=C
    B*C = D

    A^2= A; Hence A should be 1
    AB+AB = B; Here A=1; B+B = B Then B=0
    BC = D; we know that B=0 then D =0
    C= can take 0 through 9

    ABC * AB = ABCD = 100 * 10 = 1000
    =101*10 =1010
    =102*10=1020; it goes like this upto 109*10 =1090
    Final: A= 1; B=0; D=0; C= 0 through 9
    AB = 10;

    (1) AD = 10 it should be factor of ABCD not matter what the value of C
    (2)CD = 00;10;20;30;40;50;60;70;80;90
    Example ABCD= 1080 then CD =80 Not a factor of ABCD
    (3)CA=01;11;21;31;41;51;61;71;81;91;
    Example: 1080 then CA=81 not a factor of ABCD.
    Likewise we can test cases for 2 and 3 and conclude that CD and CA are not factor of ABCD.
    Hence, ans is D.
    But, is this 2 min problem... What is the correct 2 min approach...

    • After testing all the cases I would choose Ans A. None.

      (2) ABCD=1010 CD=10 10 is a factor of 1010
      (3) ABCD=1000 CA =01 1 is a factor of 1000

      So Ans A.

    • hariharakarthi, well done! You did a great job breaking down this question, and since you and iswar between you were the first to cut to the heart of the question, we're declaring the two of you "co-winners"! If you send an email to abby AT knewton.com, giving her your physical address, we'll have your T-shirt sent to you as soon as possible!

      (I'll address your concern about the "first digit" below in just a moment! Jon Myers' comment is indeed accurate. The note means that C cannot be equal to 0, so CA cannot be a factor of A,BCD, making answer choice C correct.)

  • For 1020

    AD: 10
    CD:20
    CA:21
    here only CA is not the factor

    For 1030:
    AD:10
    CD:30
    CA:31

    here
    CA,CD both are not factors.

    I am not able to come to conclusion!

  • The Answer is (A). Try two or three combination of A, BCD numbers and every answer choice is a factor of the number. Picking numbers is the easiest work around!!

  • ANS IS C
    AB=10*A+B
    ABC=100*A+10*B+C
    ABCD=1000*A+100*B+10*C+D

    AB*ABC=1000A^2 +100AB+10AC+100AB+10B^2+BC
    =ABCD=1000*A+100*B+10*C+D

    SO,A^2=A SO A=1

    100AB=100B
    10AC=10C

    SO D=100AB+10B^2+BC

    THIS IS POSSIBLE ONLY IF B=0 SO,D=0

    HENCE A=1,B=0,D=0 C CAN BE ANY DIGIT.

    FROM OPTIONS AD=10 WHICH IS ALWAYS A FACTOR OF ABCD SINCE D=0
    CD MY OR MAY NOT BE A FACTOR
    FOR EG. 20 IS A FACTOR OF 1020
    30 IS NOT A FACTOR OF 1030

    CA CANNOT BE A FACTOR OF ABCD.
    FOR EXAMPLE,
    31 IS NOT A FACTOR OF 1030

    SO ONLY CA CANT BE THE FACTOR.CD MAY BE FACTOR.
    SO C IS THE ANS

    • @iswar

      What about CA =01 - is a factor of all the integers.
      So, we can't say CA can't be a factor of ABCD. So, ANS is A. None.
      Correct me, if I am wrong.

    • Well done, as ever, Iswar! The correct answer is indeed C.

      You should check your calculations for determining the digits B, C, and D, however. hariharakarthi's calculations above are more accurate, because a number multiplied by 100 or 10 cannot be equal to a single digit. The equations describing B, C, and D should be:
      B = 2AB
      C = B^2 + AC
      D = BC

      However, the result you came to is indeed accurate: A = 1, B = D = 0, and C can be any digit.

      Since you did a good job of dealing with "which of the following CANNOT," and hariharakarthi did great work with the rest of the question up until there, we're going to declare you two "co-winners": If you write to abby AT knewton.com giving her your physical address, we'll have a shirt sent to you asap!

  • The note says the first digit of a number cannot be equal to 0, so that's why you can't use 1,000 as an example, because then CA = 01. So the answer can't be A.

  • Jon Myers-

    There's ambiguity as to whether "a number cannot start with 0" means that A cannot be zero, or that none of the answer choices can start with 0. Unfortunately, I have found that this is pretty typical of the way Knewton words its questions (though this is the first time I've seen it in math, in the past that thought has occurred to me on verbal sections). My guess (and hope) is we won't see anything this ambiguous on the actual test.

    It could be that:
    A = 1
    B = C = D = 0

    in which case the answer is A.

    Or if the question suggests that C /= 0, then it would seem that C is correct since 1010 & 1020 show cases where AD and CD can be divisors.

    My guess is this post won't win me a Knerd shirt...

  • @Disappointed
    I agree. The note says the first digit of a number cannot be equal to 0. This only holds true for whatever the numbers (ABCD,ABC, and AB) in given in the question prompt.

    I don't think the question prompt note also advises us to assume that the number in the ans choice also can't start with zero.

    Any help is appreciated.

  • The answer is C. As mentioned in the question C in CA cannot be zero. So the only possibile answer is C.

  • hariharakarthi and Disappointed, when you sense an ambiguity in a question (especially one on the GMAT!), you should pay careful attention to how keywords are used. (Disappointed, you may find that
    this resolves some of your trouble on Verbal questions as well - unfortunately, your post indeed won't win you a shirt, although it could have, had you shown your work deriving the fact that A = 1 and B = D = 0! )

    In this particular instance, note the use of the term "n-digit number" throughout the question. Both AB, ABC, and the three options AD, CD, and CA are described using these keywords, which are then referenced in the note: "the first _digit of a number_ cannot be..."

    Both on the GMAT, and (I should certainly hope) in Knewton questions, the meanings of keywords like this will not shift during the course of a question. It is not possible for the "two-digit number AB" and the "following two-digit numbers" to be subject to different rules: the note must describe both!

    (And, both in Knewton questions and the GMAT, "01" can never be described as a "two-digit integer" - there are no leading zeros in positive integers; the note just makes this explicit.)

    Again, congrats to (three-time winner!) iswar, and hariharakarthi - we hope you'll be wearing your Knerd shirts soon!

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