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# A is the set of 6-digit positive integers whose first three digits are

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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A is the set of 6-digit positive integers whose first three digits are  [#permalink]

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Updated on: 26 Feb 2019, 06:06
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35% (medium)

Question Stats:

65% (01:44) correct 35% (01:43) wrong based on 49 sessions

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A is the set of 6-digit positive integers whose first three digits are same as their last three digits, written in the same order. Which of the following numbers must be a factor of every number in the set A?

A. 6
B. 11
C. 17
D. 19
E. 23

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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Originally posted by MathRevolution on 26 Feb 2019, 00:28. Last edited by chetan2u on 26 Feb 2019, 06:06, edited 1 time in total. Corrected the Q Math Expert Joined: 02 Aug 2009 Posts: 7958 Re: A is the set of 6-digit positive integers whose first three digits are [#permalink] ### Show Tags 26 Feb 2019, 01:27 1 MathRevolution wrote: A is the set of 6-digit positive integers whose first three digits are same as their last three digits, written in the same order. Which of the following numbers must be a factor of every number in the set A? A. 7 B. 11 C. 17 D. 19 E. 23 Let the number be abcabc, where a, b and c are digits. Now an integer, abcabc, can be written as $$a*100000+b*10000+c*1000+a*100+b*10+c)=a(100000+100)+b(10000+10)+c(1000+1)=a(100100)+b(10010)+c(1001)=1001(100a+10b+c)$$ Thus, each number is a multiple of 1001, which is 11*7*13 So, each number has to be a factor of 7, 11 and 13.. Thus, both A and B can be the answer. _________________ Senior Manager Joined: 04 Aug 2010 Posts: 474 Schools: Dartmouth College Re: A is the set of 6-digit positive integers whose first three digits are [#permalink] ### Show Tags 26 Feb 2019, 06:01 The GMAT is unlikely to test divisibility by 7. An integer of the form XYZXYZ must be divisible by 7, but this issue seems irrelevant to the GMAT. For this reason, I've replaced answer choice A with the value in red: Max@Math Revolution wrote: A is the set of 6-digit positive integers whose first three digits are same as their last three digits, written in the same order. Which of the following numbers must be a factor of every number in the set A? A. 5 B. 11 C. 17 D. 19 E. 23 To determine whether an integer is divisible by 11: 1. From the left to right, sum alternating digits 2. Sum the remaining digits 3. Calculate the difference between the sums 4. If the difference is divisible by 11, so is the integer Example: 587686 1. Sum of the blue digits = 5+7+8= 20 2. Sum of the red digits = 8+6+6 = 20 3. Difference between the sums = 20-20 = 0 4. Since the difference is divisible by 11, 587686 is divisible by 11 Each integer in set A is constructed as follows: XYZXYZ 1. Sum of the blue digits = X+Z+Y 2. Sum of the red digits = Y+X+Z 3. Difference between the sums = (X+Z+Y) - (Y+X+Z) = 0 4. Since the difference is divisible by 11, XYZXYZ must be divisible by 11 _________________ GMAT and GRE Tutor Over 1800 followers GMATGuruNY@gmail.com New York, NY If you find one of my posts helpful, please take a moment to click on the "Kudos" icon. Available for tutoring in NYC and long-distance. For more information, please email me at GMATGuruNY@gmail.com. Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8011 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: A is the set of 6-digit positive integers whose first three digits are [#permalink] ### Show Tags 28 Feb 2019, 07:19 Each number n in the set A is an integer of the form “xyz,xyz”. So, n = 10^5x + 10^4y + 10^3z + 10^2x + 10y + z = 10^3(10^2x + 10y + z) + (10^2x + 10y + z ) = 1000(10^2x + 10y + z) + (102x + 10y + z ) = 1001(10^2x + 10y + z ) = 11*91(10^2x + 10y + z ) Thus, n is a multiple of 11. Therefore, B is the answer. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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Re: A is the set of 6-digit positive integers whose first three digits are   [#permalink] 28 Feb 2019, 07:19
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