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Math Expert V
Joined: 02 Sep 2009
Posts: 59712
If the difference between a two digit positive integer and its reversa  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 62% (01:53) correct 38% (02:11) wrong based on 66 sessions

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Competition Mode Question

If the difference between a two digit positive integer and its reversal is three times the sum of its digits, how many such pairs are there?

A. 1
B. 2
C. 3
D. 4
E. 5

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Re: If the difference between a two digit positive integer and its reversa  [#permalink]

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xy-yx=3(x+y)
10x+y-10y-x=3x+3y
x=2y

4digits(x={1,2,3,4};y={2,4,6,8}) satisfies above equation

OA:D
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Re: If the difference between a two digit positive integer and its reversa  [#permalink]

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2
If the difference between a two digit positive integer and its reversal is three times the sum of its digits, how many such pairs are there?

A. 1
B. 2
C. 3
D. 4
E. 5

Let the two digits number be 10a + b
Reverse is 10b + a

Now, 10a + b - 10b - a = 3(a + b)
9a - 9b = 3a + 3b
6a = 12b
a = 2b

'a' can have only four values for which 'b' would have four values. The numbers are:
a = 1, b = 2
a = 2, b = 4
a = 3, b = 6
a = 4, b = 8

Hence only four numbers are possible.

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Re: If the difference between a two digit positive integer and its reversa  [#permalink]

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1
10a+b-10b-a=3(a+b)

3(a-b)=(a+b)
a=2b
possible pairs ( 1,2) ( 2,4) (3,6) (4,8)
IMO D ; 4

If the difference between a two digit positive integer and its reversal is three times the sum of its digits, how many such pairs are there?

A. 1
B. 2
C. 3
D. 4
E. 5
Director  P
Joined: 24 Nov 2016
Posts: 963
Location: United States
Re: If the difference between a two digit positive integer and its reversa  [#permalink]

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1
Quote:
If the difference between a two digit positive integer and its reversal is three times the sum of its digits, how many such pairs are there?

A. 1
B. 2
C. 3
D. 4
E. 5

10A+B-(10B+A)=3(A+B)
9(A-B)=3(A+B)
A=2B
0<A≤9
A={2,4,6,8}; B={1,2,4,3}
(A,B)={2,1;4,2;6,4;8,3}: 4 pairs

Ans (D)
Senior Manager  P
Joined: 22 Feb 2018
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Re: If the difference between a two digit positive integer and its reversa  [#permalink]

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2
Imo. D

If the difference between a two digit positive integer and its reversal is three times the sum of its digits, how many such pairs are there?
10a+b - (10b+a) = 3 (a+b)
9a-9b = 3a+3b
6a=12b
a/b=2/1
Let's check the respective numbers
21-12 = 3 *3
42-24 = 3 * 6
63-36 = 3 * 9
84-48 = 3 * 12

Sp, total 4 such pairs are possible.

Hence, D.
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Re: If the difference between a two digit positive integer and its reversa  [#permalink]

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1
Let the two digit number be m = xy = 10x + y
And number obtained by reversing be n = yx = 10y + x

Difference = 10x + y - (10y + x) = 9x - 9y = 9(x - y)

Given, 9(x - y) = 3(x + y)
—> 3(x - y) = x + y
—> 2x = 4y
—> x = 2y
—> Possible values = 12, 24, 36, 48
—> Possible pairs of 2 digit number and its reversal = {(12,21). (24,42), (36,63), (48,84)}
—> 4 pairs

IMO Option D

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VP  P
Joined: 07 Dec 2014
Posts: 1217
Re: If the difference between a two digit positive integer and its reversa  [#permalink]

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1
10x+y-(10y+x)=3(x+y)→
x=2y
so 4 possible pairs are:
21&12
42&24
63&36
84&48
Director  P
Joined: 18 May 2019
Posts: 549
Re: If the difference between a two digit positive integer and its reversa  [#permalink]

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1
Let the two digit positive number =ab
then its reverse is ba
ab-ba=3(a+b)
but ab=10a+b and ba=10b+a
10a+b-(10b+a)=3a+3b
9a-9b=3a+3b
6a-12b=0
6(a-2b)=0
This implies a=2b
since when b=0 a=0, b cannot be 0
only four other possible values exist which yield unique value for a, since the largest unit digit that a can take is 8 based on the given condition of a=2b.
The set of possible values of b={1, 2, 3, 4}

The answer is, therefore, option D in my view.
Intern  Joined: 26 Nov 2017
Posts: 1
Re: If the difference between a two digit positive integer and its reversa  [#permalink]

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1

x=2 y=1
x=4 y=2
x=6 y=3
x=8 y=4
Target Test Prep Representative V
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Re: If the difference between a two digit positive integer and its reversa  [#permalink]

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1
Bunuel wrote:

Competition Mode Question

If the difference between a two digit positive integer and its reversal is three times the sum of its digits, how many such pairs are there?

A. 1
B. 2
C. 3
D. 4
E. 5

Are You Up For the Challenge: 700 Level Questions

We can let a = tens digit and b = ones digit, and create the equation:

10a + b - (10b + a) = 3(a + b)

10a + b - 10b - a = 3a + 3b

6a = 12b

a = 2b

We see that a can be 2, 4, 6, or 8, so there are 4 such pairs.

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