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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 AA [#permalink] New post 06 Aug 2012, 09:57
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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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Re: AB + CD = AAA, where AB and CD are two-digit numbers and AA [#permalink] New post 06 Aug 2012, 10:06
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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] New post 18 Aug 2012, 00:24
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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] New post 16 Aug 2012, 10:11
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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Re: AB + CD = AAA, where AB and CD are two-digit numbers and AA [#permalink] New post 16 Aug 2012, 10:14
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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] New post 17 Aug 2012, 21:10
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] New post 10 Nov 2012, 05:47
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] New post 25 Feb 2014, 19:43
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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] 25 Feb 2014, 19:43
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