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

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

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New post Updated on: 26 Feb 2019, 06:06
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Question Stats:

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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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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
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Re: A is the set of 6-digit positive integers whose first three digits are  [#permalink]

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New post 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.
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Re: A is the set of 6-digit positive integers whose first three digits are  [#permalink]

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New post 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


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Re: A is the set of 6-digit positive integers whose first three digits are  [#permalink]

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New post 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.
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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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