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Find a positive integer containing all ten digits: $0,1,2,3,4,5,6,7,8,9$ that is a multiple of $126$

I don't really know where to start. I guess I could find the prime factorization of $126$, which is $2*3^2*7$, but I don't know how that helps. The only thing I can think of is just multiplying the number, but that would take forever.

suomynonA
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    Hint: the sum of those digits is $45$ which is divisible by $9$...hence any $10$ digit number made by permuting them will be divisible by $9$. It's easy to make the number divisible by $2$, so your only problem is $7$. – lulu Sep 11 '16 at 19:16

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Since a number containing all the digits $\ 0,1,2,\cdots,9\ $ exactly once is divisble by $9$, we only have to make sure that the number is even and divisible by $7$.

Since the numbers $\ 91,56,203,84,7\ $ are divisible by $7$, the number

$$9156203784$$ must be divisible by $7$ and hence must be divisble by $126$.

Peter
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  • How do you find the numbers $91,56,203,84$? I know why you use them, but I don't know how to get them from $0,1,2,3,4,5,6,7,8,9$ – suomynonA Sep 11 '16 at 19:21
  • I started with $91$ and $56$ and had the luck that the rest worked :) – Peter Sep 11 '16 at 19:22
  • how do i find it if I'm out of luck? ;) – suomynonA Sep 11 '16 at 19:26
  • Start with , for example, $1234567890$ and begin to permut the last two digits, the last three digits and so on You will be successful with $1234567908$. – Peter Sep 11 '16 at 19:50
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You said a positive integer, not the smaller positive integer. You said also that is must contain each integer, not that it must contain each integer once.

Then I have a solution in the pocket

Take $1234567890000$ , compute its modulo $126$ and the difference of the latter with $126$ and you find $108$. Then just add them and deliver your result : $1234567890108$

If it is not a valid answer, I'm sorry for the trick. Let's always be careful with the statements :)

  • I guess I can do that without a calculator, but it might be hard – suomynonA Sep 12 '16 at 04:51
  • @Anonymous : I did the division with my phone, with the calc windows utility and with my old pen : all render 108 for 1234567890000-floor(1234567890000/126) ... It is probably the solution expected by the teacher :) –  Sep 18 '16 at 14:44