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# If a three-digit positive integer has its digits reversed, the

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Math Expert
Joined: 02 Sep 2009
Posts: 60644
If a three-digit positive integer has its digits reversed, the  [#permalink]

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07 Sep 2017, 04:57
3
40
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Difficulty:

95% (hard)

Question Stats:

32% (02:27) correct 68% (02:29) wrong based on 374 sessions

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Fresh GMAT Club Tests' Challenge Question:

If a three-digit positive integer has its digits reversed, the resulting three-digit positive integer is less than the original integer by 297. How many such pairs are possible?

A. 3
B. 6
C. 7
D. 60
E. 70

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Senior Manager
Joined: 29 Jun 2017
Posts: 417
GPA: 4
WE: Engineering (Transportation)
If a three-digit positive integer has its digits reversed, the  [#permalink]

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Updated on: 17 Dec 2017, 03:49
11
8
IMO D

abc -cba =297
100a+10b+c - (100c+10b+a) = 297
100(a-c) +c-a = 297
99(a-c) =297
a-c =3
b can take 0-9 => 10 values
but c can not be 0 since reverse number is also 3 digit number
c max = 6 because 6+3 =9 =a max
so c can take 1,2,3,4,5,6 => 6 values
Hence total numbers possible
6x10=60

Originally posted by sahilvijay on 07 Sep 2017, 05:17.
Last edited by sahilvijay on 17 Dec 2017, 03:49, edited 2 times in total.
##### General Discussion
Intern
Joined: 04 Oct 2016
Posts: 7
Re: If a three-digit positive integer has its digits reversed, the  [#permalink]

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07 Sep 2017, 05:55
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

If a three-digit positive integer has its digits reversed, the resulting three-digit positive integer is less than the original integer by 297. How many such pairs are possible?

A. 3
B. 6
C. 7
D. 60
E. 70

100x+10y+z-100z+10y+x=297
99(x-z)=297
x-z = 3
there are 6 possible numbers for x. for example: 9y6-6y9=297 and so on till x=4
as for y. there are 10 possibilities: from 0 to 9
so 6*10=60 pairs possible.
Senior Manager
Joined: 12 Dec 2015
Posts: 474
Re: If a three-digit positive integer has its digits reversed, the  [#permalink]

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07 Sep 2017, 06:39
1
Ans: D

N=100x+10y+z
=> (100x+10y+z) - (100x+10y+z) =297
=> x-z =3
(1) x <=9, z <=6 <=> Total 6 pairs z=1,2,3,4,5 & 6
(2) y can be 0 to 9(Total 10)

Total possibilities = 6*10=60
Intern
Joined: 26 Sep 2017
Posts: 12
Re: If a three-digit positive integer has its digits reversed, the  [#permalink]

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07 Oct 2017, 04:44
sahilvijay wrote:
IMO D

abc -cba =297
100a+10b+c - (100c+10b+a) = 297
100(a-c) +c-a = 297
99(a-c) =297
a-c =3
b can take 0-9 => 10 values
but c can not be 0 since reverse number is also 3 digit humber
so c can take 1,2,3,4,5,6 => 6 values
Hence total numbers possible
6x10=60

b*c= 10 * 6= 60
so what about a? where have it gone?
Senior Manager
Joined: 29 Jun 2017
Posts: 417
GPA: 4
WE: Engineering (Transportation)
Re: If a three-digit positive integer has its digits reversed, the  [#permalink]

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10 Oct 2017, 01:00
lor12345 wrote:
sahilvijay wrote:
IMO D

abc -cba =297
100a+10b+c - (100c+10b+a) = 297
100(a-c) +c-a = 297
99(a-c) =297
a-c =3
b can take 0-9 => 10 values
but c can not be 0 since reverse number is also 3 digit humber
so c can take 1,2,3,4,5,6 => 6 values
Hence total numbers possible
6x10=60

b*c= 10 * 6= 60
so what about a? where have it gone?

No need to consider a - above solution is self explanatory
Intern
Joined: 19 Sep 2011
Posts: 25
Re: If a three-digit positive integer has its digits reversed, the  [#permalink]

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11 Oct 2017, 17:56
1
Possible combinations are 4b1, 5b2, 6b3,7b4,8b5, 9b6
B can be any dogit from 0-9 , hence 10 possiblities. and the above combination are only 6 , therefore answer is 6*10 = 60
VP
Joined: 07 Dec 2014
Posts: 1229
If a three-digit positive integer has its digits reversed, the  [#permalink]

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18 Sep 2018, 20:59
3
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

If a three-digit positive integer has its digits reversed, the resulting three-digit positive integer is less than the original integer by 297. How many such pairs are possible?

A. 3
B. 6
C. 7
D. 60
E. 70

xyz-zyx=297
297/99=3=x-z
6 possible x-z combinations: 9-6,8-5,7-4,6-3,5-2,4-1
10 possible y values: 0-9
6*10=60 possible pairs
D
Math Expert
Joined: 02 Sep 2009
Posts: 60644
Re: If a three-digit positive integer has its digits reversed, the  [#permalink]

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24 Dec 2018, 04:56
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

If a three-digit positive integer has its digits reversed, the resulting three-digit positive integer is less than the original integer by 297. How many such pairs are possible?

A. 3
B. 6
C. 7
D. 60
E. 70

_________________
SVP
Joined: 03 Jun 2019
Posts: 1942
Location: India
GMAT 1: 690 Q50 V34
Re: If a three-digit positive integer has its digits reversed, the  [#permalink]

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08 Oct 2019, 01:07
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

If a three-digit positive integer has its digits reversed, the resulting three-digit positive integer is less than the original integer by 297. How many such pairs are possible?

A. 3
B. 6
C. 7
D. 60
E. 70

Asked: If a three-digit positive integer has its digits reversed, the resulting three-digit positive integer is less than the original integer by 297. How many such pairs are possible?

Let the 3-digit positive integer be of the form xyz

(100x + 10y + z) - (100z + 10y + x) = 99(x-z) = 297
x -z = 3

Since x & z are not 0

(x,z) = {(4,1),(5,2),(6,3),(7,4),(8,5),(9,6)} = 6 cases
y can take 10 values

Total such numbers = 6*10 = 60

IMO D
VP
Joined: 24 Nov 2016
Posts: 1100
Location: United States
Re: If a three-digit positive integer has its digits reversed, the  [#permalink]

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18 Nov 2019, 06:07
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

If a three-digit positive integer has its digits reversed, the resulting three-digit positive integer is less than the original integer by 297. How many such pairs are possible?

A. 3
B. 6
C. 7
D. 60
E. 70

$$ABC-CBA=297:100a+10b+c-100c-10b-a=297, 99(a-c)=297, (a-c)=3$$
$$(A,C)=(9,6;8,5;7,4;6,3;5,2;4,1)…(A,C)≠(3,0):C>0$$
$$(A,C)=6.pairs…B=(0,1,2,3,4,5,6,7,8,9)=10$$
$$Total.cases:6*10=60$$

Ans (D)
Re: If a three-digit positive integer has its digits reversed, the   [#permalink] 18 Nov 2019, 06:07
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