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Could you show me that how can I solve this as fast as possible, please?

$$ ABCDEF \, \land \, DEFABC \; \large{ \in \, \mathbb{N^{+}} } $$ $$ ABCDEF \, = \, 6 \times (DEFABC) \\ $$ $$ (A+B+C+D+E+F)=\; ? $$

Thank you very much... :)

2 Answers 2

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DEFABC = 142857 obviously here.

More systematically though, let the sum be X. The equation shows that X is divisible 3, so applying it again shows that X is divisible by 9. Assuming that ABCDEF are all distinct, we have to choose 4 digits to reject. Overall sum of 0-9 is 45, so it must be less than that. Sum of rejects being 9 would make it hard to construct DEFABC as 0,1 would have to be rejected. Sum of rejects being 27 would have required us to reject 9, making ABCDEF hard to construct. So sum of rejects must be 18, thus X is 27.

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    Funny, the period of $1/7$.2012-04-17
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    @Wonder: How could you go that far, could you show me that, please? I did and answered it wrong (I found 21.)...2012-04-17
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    @draks: What does that "the period of 1/7" mean?2012-04-17
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    @Wonder: Thank you, once more, I need to look at the problem more carefully now... :)2012-04-17
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    @KerimAtasoy: Compare the decimal expansions of 1/7 and 6/7.2012-04-17
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    @KerimAtasoy, it is just a common sort of question that keeps coming up because 142857 gets its digits permuted when multiplied by any number from 1 to 6. I hope the more detailed explanation I gave is of help in how to go about it systematically.2012-04-17
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    Yes, sir, thank you very much for your all help... :)2012-04-17
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Here's another solution: we immediately see that $D=1$, since otherwise $6\times DEFABC$ would have to have more than seven digits. Now write: $X=ABC$, $Y=DEF$. The equation becomes: $1000X+Y=6(1000Y+X)$ which can be rewritten as $994X=5999Y$. Now, since $994$ is divisible by the primes $2$ and $71$, and $5999$ isn't, $Y$ will have to be divisible by $2$ and $71$. But then $Y$ is divisible by $142$ and since the first digit of $Y$ is $1$, it follows that $Y=142$. Now $X=857$ follows easily and we arrive at the same (unique) solution.