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# The “hash” of a three-digit integer with three distinct integers is de

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Math Expert
Joined: 02 Sep 2009
Posts: 58320
The “hash” of a three-digit integer with three distinct integers is de  [#permalink]

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05 Feb 2019, 02:57
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Difficulty:

25% (medium)

Question Stats:

71% (01:30) correct 29% (02:02) wrong based on 55 sessions

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The “hash” of a three-digit integer with three distinct integers is defined as the result of interchanging its units and hundreds digits. The absolute value of the difference between a three-digit integer and its hash must be divisible by

A. 9
B. 7
C. 5
D. 4
E. 2

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Intern
Joined: 28 Sep 2018
Posts: 29
Re: The “hash” of a three-digit integer with three distinct integers is de  [#permalink]

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05 Feb 2019, 03:14
Let the three digit number = ABC

Hence the "hash" will = CBA

Now a three digit number can be written as 100A+10B+C

Hence the difference will be

100A+10B+C - (100C+10B+A)

Upon simplification we can write the above as

100(A-C) + (C-A)

100A-100C+C-A = 99(A-B)

Cleary 9 is divisible

Alternate method (not recommended)

Plug in values for the three digit number.

Eg-
302-203= 99

99/9 gives an integer value

(A)

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Director
Joined: 09 Mar 2018
Posts: 997
Location: India
Re: The “hash” of a three-digit integer with three distinct integers is de  [#permalink]

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05 Feb 2019, 03:16
Bunuel wrote:
The “hash” of a three-digit integer with three distinct integers is defined as the result of interchanging its units and hundreds digits. The absolute value of the difference between a three-digit integer and its hash must be divisible by

A. 9
B. 7
C. 5
D. 4
E. 2

IMO A

Let 3 digit number be 100x + 10y + z

hash = 100z +10y + x

difference between a three-digit integer and its hash = 100x + 10 y + z - 100z - 10y -x
= 99x -99z
= 99 (x-z)

Divisible by 9
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Senior Manager
Joined: 10 Jan 2013
Posts: 294
Location: India
Concentration: General Management, Strategy
GPA: 3.95
Re: The “hash” of a three-digit integer with three distinct integers is de  [#permalink]

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18 May 2019, 01:17
Bunuel wrote:
The “hash” of a three-digit integer with three distinct integers is defined as the result of interchanging its units and hundreds digits. The absolute value of the difference between a three-digit integer and its hash must be divisible by

A. 9
B. 7
C. 5
D. 4
E. 2

My soln

let the number be abc

also expressed as = 100a + 10b + c

given the hash (#)
100c + 10b + a

as per q
100c + 10b + a - 100a - 10b - c

99c - 99a

99(c-a)

only A works
Re: The “hash” of a three-digit integer with three distinct integers is de   [#permalink] 18 May 2019, 01:17
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