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AB + CD = AAA, where AB and CD are two-digit numbers and AA

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AB + CD = AAA, where AB and CD are two-digit numbers and AAA is a three digit number; A, B, C, and D are distinct positive integers. In the addition problem above, what is the value of C?

(A) 1
(B) 3
(C) 7
(D) 9
(E) Cannot be determined
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navigator123 wrote:
AB + CD = AAA, where AB and CD are two-digit numbers and AAA is a three digit number; A, B, C, and D are distinct positive integers. In the addition problem above, what is the value of C?

(A) 1
(B) 3
(C) 7
(D) 9
(E) Cannot be determined


Since AB and CD are two-digit integers, their sum can give us only one three digit integer of a kind of AAA: 111.

So, A=1 and we have 1B+CD=111

Now, C can not be less than 9, because no two-digit integer with first digit 1 (1B<20) can be added to two-digit integer less than 90, so that to have the sum 111 (if CD<90, so if C<9, CD+1B<111).

Hence C=9.

Answer: D.
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Re: AB + CD = AAA, where AB and CD are two-digit numbers and AA [#permalink]

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ratinarace wrote:
Hi Bunuel

How did u check that there is only one such number as AAA when AB and CD are added together..Please can you explain the logic behind it?


AAA is a 3-digit number with all 3 digits alike, so it could be: 111, 222, 333, ..., 999. But the sum of any two 2-digit numbers cannot be more than 99+99=198, so AB+CD can only be 111.
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Re: AB + CD = AAA, where AB and CD are two-digit numbers and AA [#permalink]

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New post 16 Aug 2012, 11:11
Hi Bunuel,

Doesn't 82+19=111. But you say no two digit number with 1st digit as 1 can be added to a number less than 90 to get a sum of 111. Can you elaborate??

One more thing is only 111 fits into this scenario as it is the sum of two digit numbers which cannot exceed 188 right??
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New post 16 Aug 2012, 11:14
rajathpanta wrote:
Hi Bunuel,

Doesn't 82+19=111. But you say no two digit number with 1st digit as 1 can be added to a number less than 90 to get a sum of 111. Can you elaborate??

One more thing is only 111 fits into this scenario as it is the sum of two digit numbers which cannot exceed 188 right??


82+19=101, not 111.

As for 111: since AB and CD are two-digit integers, their sum can give us only one three digit integer of a kind of AAA: 111.
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Re: AB + CD = AAA, where AB and CD are two-digit numbers and AA [#permalink]

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New post 17 Aug 2012, 22:10
Hi Bunuel

How did u check that there is only one such number as AAA when AB and CD are added together..Please can you explain the logic behind it?

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Re: AB + CD = AAA, where AB and CD are two-digit numbers and AA [#permalink]

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New post 10 Nov 2012, 06:47
Bunnuel, is there any algebraic approach to this?
Please help out
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Re: AB + CD = AAA, where AB and CD are two-digit numbers and AA [#permalink]

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New post 25 Feb 2014, 20:43
AB+CD=AAA

10A+B+10C+D=100A+10A+A
10C+B+D=101A
(10C+B+D)/101=A

Just need to realize that A,B,C,D are integers that cannot exceed 9, as they are single digits

Keeping this in mind, C is at most 9, so (90+B+D)/101=A -->B+D=11 and A=1

For A = 2, it becomes obvious that either B, C, or D will have to exceed 9 and is thus not possible.

So C=9
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Re: AB + CD = AAA, where AB and CD are two-digit numbers and AA [#permalink]

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Re: AB + CD = AAA, where AB and CD are two-digit numbers and AA [#permalink]

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New post 06 Oct 2015, 03:19
the greatest 2 digit number is 99 so 99+99 =198...
thats shows that AAA can not be 222 or 333 or ...... 999, it has to be 111....
now that aaa is 111,ab+cd=aaa ...
a bit of logic, a=1 and c=9...
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Re: AB + CD = AAA, where AB and CD are two-digit numbers and AA [#permalink]

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New post 18 Mar 2017, 10:29
we can get there by trying numbers:
AB
CD
AAA
first what come to my mind is that A can be 1, hence B+D=11, then 1(A) + 9 (C)=10 + 1
Answer is D

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AB + CD = AAA, where AB and CD are two-digit numbers and AA [#permalink]

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New post 20 Jul 2017, 03:45
Please ignore

One thing i am clear that the no has to be 111. But why is 24+87=111 wrong?

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New post 20 Jul 2017, 05:50
gps5441 wrote:
Please ignore

One thing i am clear that the no has to be 111. But why is 24+87=111 wrong?


AB + CD = AAA.

The tens digit in AB and the digits in AAA are the same. So, A = 1. All this is explained here: https://gmatclub.com/forum/ab-cd-aaa-wh ... l#p1110751
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Re: AB + CD = AAA, where AB and CD are two-digit numbers and AA [#permalink]

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New post 21 Jul 2017, 05:11
Bunuel Thanks

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Re: AB + CD = AAA, where AB and CD are two-digit numbers and AA   [#permalink] 21 Jul 2017, 05:11
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