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The "reflection" of a positive integer is obtained by reversing its di

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Bunuel
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The "reflection" of a positive integer is obtained by reversing its di [#permalink] New post   07 Feb 2019, 00:56
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The "reflection" of a positive integer is obtained by reversing its digits. For example 321 is the reflection of 123 The difference between a five-digit integer and its reflection must be divisible by which of the following?

A. 2
B. 4
C. 5
D. 6
E. 9
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E
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chandrag
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Re: The "reflection" of a positive integer is obtained by reversing its di [#permalink] New post   09 Feb 2019, 10:55
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Number = abcde = 10^4 ( a ) + 10^3 (b) + 10^2 (c) + 10 (d) + e
Reflection = edcba = 10^4 ( e ) + 10^3 (d) + 10^2 (c) + 10 (b) + a


Number - Reflection = 10^4 ( a - e ) + 10^3 ( b -d ) + 10 (d- b) + e - a

= (a-e) ( 10 ^ 4 - 1) + (b-d) ( 10^3 - 10)
= (a-e) (99999) + (b-d) (990)
= 9* (something)

Hence the answer.
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Re: The "reflection" of a positive integer is obtained by reversing its di [#permalink] New post   07 Feb 2019, 01:07
Bunuel wrote:
The "reflection" of a positive integer is obtained by reversing its digits. For example 321 is the reflection of 123 The difference between a five-digit integer and its reflection must be divisible by which of the following?

A. 2
B. 4
C. 5
D. 6
E. 9


eg = 12345 and reflection 54321 , 45789 , reflection 98754

testing the values , difference would always be divisible by 9
IMO E
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Re: The "reflection" of a positive integer is obtained by reversing its di [#permalink] New post   09 Feb 2019, 11:19
Bunuel wrote:
The "reflection" of a positive integer is obtained by reversing its digits. For example 321 is the reflection of 123 The difference between a five-digit integer and its reflection must be divisible by which of the following?

A. 2
B. 4
C. 5
D. 6
E. 9


when the number is taken as 54321, reflection will be 12345, difference will be / by 2 4 and 9

when the number is taken as 76432, reflection will be 23467, difference will be / 9

Answer E
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Re: The "reflection" of a positive integer is obtained by reversing its di [#permalink] New post   12 Jan 2020, 02:32
Bunuel wrote:
The "reflection" of a positive integer is obtained by reversing its digits. For example 321 is the reflection of 123 The difference between a five-digit integer and its reflection must be divisible by which of the following?

A. 2
B. 4
C. 5
D. 6
E. 9


What is the mathematical concept behind this ?
I, like everyone else, did the examples.
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indugopal1991
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Re: The "reflection" of a positive integer is obtained by reversing its di [#permalink] New post   23 Jun 2020, 12:00
The difference should be always a even number, and as it is divided by 9 so it should be divided by 6 as well, so out of "2", "6" and "9", why are we always taking "9" only?

**Need Explanation
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The "reflection" of a positive integer is obtained by reversing its di [#permalink] New post   Updated on: 23 Jun 2020, 12:38
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indugopal1991 wrote:
The difference should be always a even number, and as it is divided by 9 so it should be divided by 6 as well, so out of "2", "6" and "9", why are we always taking "9" only?

**Need Explanation


Dear indugopal1991 ,

Take example :1. 12345 its reflection is 54321 is divisible2,6, 9.
2. 45789 its reflection 98754 is only divisible 9
3. 34567 its reflection 76543

For all 5 digit number reflection, it is only divisible by 9. Sometime may or may not be it is divisible by any other number apart from 9

Regards,
Rajat Chopra

Originally posted by rajatchopra1994 on 23 Jun 2020, 12:22.
Last edited by rajatchopra1994 on 23 Jun 2020, 12:38, edited 1 time in total.
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indugopal1991
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Re: The "reflection" of a positive integer is obtained by reversing its di [#permalink] New post   23 Jun 2020, 12:32
Dear Raj, for the first example, 54321-12345=41976, which is divisible by 2, 6 and 9.

But for the second value 45679 it will not work.
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Re: The "reflection" of a positive integer is obtained by reversing its di [#permalink] New post   23 Jun 2020, 12:54
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antartican wrote:
Bunuel wrote:
The "reflection" of a positive integer is obtained by reversing its digits. For example 321 is the reflection of 123 The difference between a five-digit integer and its reflection must be divisible by which of the following?

A. 2
B. 4
C. 5
D. 6
E. 9


What is the mathematical concept behind this ?
I, like everyone else, did the examples.

It is like this:

Let the 5 digit number look like ABCDE, where each letter signifies the digit in each of the 5 places
So its reflection is EDCBA

Value of ABCDE = \(10^4A + 10^3B + 10^2C + 10D + E\)
Value of EDCBA = \(10^4E + 10^3D + 10^2C + 10B + A\)
Difference = \((10^4-1)A + (10^3-10)B + (10-10^3)D + (1-10^4)E\)
= \(9999A + 990B - 990D - 9999E \)
= \(9(1111A + 110B - 110D - 1111E)\)

So it is divisible by 9

Answer: E
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Re: The "reflection" of a positive integer is obtained by reversing its di [#permalink] New post   29 Jun 2020, 06:39
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Bunuel wrote:
The "reflection" of a positive integer is obtained by reversing its digits. For example 321 is the reflection of 123 The difference between a five-digit integer and its reflection must be divisible by which of the following?

A. 2
B. 4
C. 5
D. 6
E. 9


Solution:

We can let 12,345 be the 5-digit integer, so its reflection is 54,321 and their difference is 54,321 - 12,345 = 41,976. At this point, we see that we can only eliminate choice C as the correct answer.

Let’s use another 5-digit integer, say 12,344. Its reflection is 44,321 and their difference is 44,321 - 12,344 = 31,977. We see that this number can be divisible only by 9, so choice E must be the correct answer.

Answer: E
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Re: The "reflection" of a positive integer is obtained by reversing its di [#permalink] New post   09 Sep 2022, 22:22
Let no.: abcde, and its mirror is edcba

abcde - edcba:

If b>d [(e+10-a) (d-1+10-b) (c-1+10-c) (b-1-d) (a-e)]
Sum equals to 27 (divisible by 9)

If b<d [(e+10-a) (d-1-b) (c-c) (b+10-d) (a-1-e)
Sum equals to 18 (divisible by 9)

Hence must be multiplied by 9

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Re: The "reflection" of a positive integer is obtained by reversing its di [#permalink]
09 Sep 2022, 22:22
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