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Math Help - Help to Find the smallest natural number with these features:

  1. #1
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    Help to Find the smallest natural number with these features:

    I understand that I must learn math myself, but this exercise isn't as easy as I thought, so I need a help or any advices how to solve it.

    So, we have to find the smallest natural [ ] number with these features:

    1) The first digit is 1.

    2) If we moved 1 to the end of the number, we would get a three times bigger/greater number.


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  2. #2
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    Re: Help to Find the smallest natural number with these features:

    the first digit is 1,
    since 3 times the number ends with 1 the original number should end with 7
    also the number is divisible by 3.
    so you have to check numbers such that : the f number starts with one, ends with seven and divisable by 3
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    Re: Help to Find the smallest natural number with these features:

    Also,
    the second digit can not be 9,8,7,6,5
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    Re: Help to Find the smallest natural number with these features:

    Why the original number should end with 7?
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    Re: Help to Find the smallest natural number with these features:

    since when you multiply it by three it should end with 1. If you tried other possibility you would never get 1.
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    Re: Help to Find the smallest natural number with these features:

    I know that number can be divisable by 3 if the sum of number's digits is divisable by 3. I tried to find an answer, but I think that I am doing wrong something. I think that number is really big :/ I checked some four-figure numbers, but all of them were wrong. Hmmm :/
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    Re: Help to Find the smallest natural number with these features:

    So how to solve it?
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    Re: Help to Find the smallest natural number with these features:

    Hello, Sott!

    Find the smallest natural number with these features:
    . . 1) The first digit is 1.
    . . 2) If we moved 1 to the end of the number, we would get a three times greater number.


    We have: . \begin{array}{cccccc}1&A&B&C&D & E \\ \times &&&&& 3 \\ \hline A&B&C&D& E & 1 \end{array}


    We see that E = 7.

    We have: . \begin{array}{cccccc}1&A&B&C&D&7 \\ \times &&&&& 3 \\ \hline A&B&C&D& 7 & 1 \end{array}


    We see that D = 5.

    We have: . \begin{array}{cccccc}1&A&B&C&5&7 \\ \times &&&&& 3 \\ \hline A&B&C&5& 7 & 1 \end{array}


    We see that C = 8.

    We have: . \begin{array}{cccccc}1&A&B&8&5&7 \\ \times &&&&& 3 \\ \hline A&B&8&5& 7 & 1 \end{array}


    We see that B = 2.

    We have: . \begin{array}{cccccc}1&A&2&8&5&7 \\ \times &&&&& 3 \\ \hline A&2&8&5& 7 & 1 \end{array}


    We see that A = 4.

    We have: . \begin{array}{cccccc}1&4&2&8&5&7 \\ \times &&&&& 3 \\ \hline 4&2&8&5& 7 & 1 \end{array}


    The smallest such number is 142857.
    Thanks from Sott
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    Re: Help to Find the smallest natural number with these features:

    WOW! Awesome Soroban! Thank you very much!!!
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    Re: Help to Find the smallest natural number with these features:

    Quote Originally Posted by Soroban View Post
    Hello, Sott!



    We have: . \begin{array}{cccccc}1&A&B&C&D & E \\ \times &&&&& 3 \\ \hline A&B&C&D& E & 1 \end{array}


    We see that E = 7.

    We have: . \begin{array}{cccccc}1&A&B&C&D&7 \\ \times &&&&& 3 \\ \hline A&B&C&D& 7 & 1 \end{array}


    We see that D = 5.

    We have: . \begin{array}{cccccc}1&A&B&C&5&7 \\ \times &&&&& 3 \\ \hline A&B&C&5& 7 & 1 \end{array}


    We see that C = 8.

    We have: . \begin{array}{cccccc}1&A&B&8&5&7 \\ \times &&&&& 3 \\ \hline A&B&8&5& 7 & 1 \end{array}


    We see that B = 2.

    We have: . \begin{array}{cccccc}1&A&2&8&5&7 \\ \times &&&&& 3 \\ \hline A&2&8&5& 7 & 1 \end{array}


    We see that A = 4.

    We have: . \begin{array}{cccccc}1&4&2&8&5&7 \\ \times &&&&& 3 \\ \hline 4&2&8&5& 7 & 1 \end{array}


    The smallest such number is 142857.
    Really nice work, but I think you have first to check it for the 3,4 and 5 digits
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