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I'm not very good at maths, but I need help with a question. This one appeared in an aptitude test.

Consider the sum: $ABC + DEF + GHI = JJJ$ .If different letters represent different digits, and there are no leading zeros, what does $J$ represent?

The solution says that $J=9$, but I don't understand how.

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    The point is that there are ten letters, and that each one represent a different cipher. So, one point is to try all possibilities, and other to consider that A, D, G and J are no zero because leading coefficients are not zero and some equation that comes from the sum.2011-09-04
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    Thanks for commenting. Yes I got the bit about A, D, G and J being non zero. But how do I go about testing for all possible cases here in a time limited environment (an aptitude test) for problems of this nature. As I said, I'm not really a math expert, but I can grasp the logic. Do you have any way to find this quickly - a "trick" ?2011-09-04
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    ABC+DEF+GHI=JJJ 140+273+586=9992012-07-05
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    @Mukesh: The question didn't ask for *a* solution, but rather *why* $J=9$.2012-07-05
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    @Asaf Karagila: by trial and error i got this2012-07-06

2 Answers 2

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Consider the sum modulo 9 ("casting out nines") so that a number becomes equal to the sum of its digits.

Then $A+B+C+D+E+F+G+H+I+J = J+J+J+J$ with the first sum being (0+1+2+3+4+5+6+7+8+9)=45=0 modulo 9. This forces $4J$ and thus $J$ to be divisible by 9.

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    +1,this is a useful approach,and it also proves that $J=9$ is the only solution for this problem.2011-09-04
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    That was elegant, though I had to read it 4 times :)2011-09-04
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    It seems like a difficult problem for a general-purpose aptitude test. What test did it come from?2011-09-04
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    @yati:It doesn't matter as long as you understand it correctly:)2011-09-04
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    @zyx Accenture :)2011-09-04
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    @FoolForMath yes, exactly :)2011-09-04
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    @zyx:It's a common aptitude problem in an [interview](http://www.google.co.in/#sclient=psy&hl=en&site=&source=hp&q=Consider+the+sum%3A+ABC%2BDEF%2BGHI%3DJJJ+.If+different+letters+represent+different+digits%2C+and+there+are+no+leading+zeros%2C+what+does+J+represent%3F&pbx=1&oq=Consider+the+sum:+ABC%2BDEF%2BGHI%3DJJJ+.If+different+letters+represent+different+digits%2C+and+there+are+no+leading+zeros%2C+what+does+J+represent%3F&aq=f&aqi=&aql=&gs_sm=e&gs_upl=833l833l0l1894l1l0l0l0l0l0l0l0ll0l0&bav=on.2,or.r_gc.r_pw.&fp=cc63231dd3ba60f6&biw=1360&bih=677).2011-09-04
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One possible way to solve this is to try and compute the integer partitions of the given options,this should not take much time as you only need to check for a sum using two and three unique digits.

Using this same idea for the option $9$, two possible partitions are $\{4,3,2\}\{7,1,1\}$ notice here there is two $1$ in the second partition,so for atleast one row sum we need a $19$ to carry up that deficit $1$ implicitly,hence $\{2,8,9\}$.

Hence we can conclude one possible solution as: $$402 +318+279 = 999$$ You can see this problem could have multiple possible solution for the sum to be all $9$'s.