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Math Help - show 3|n iff 3|s

  1. #1
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    Question show 3|n iff 3|s

    hey! the question is as follows:

    Show that 3|n if and only if 3|s, where s is the sum of the digits of n (expressed as usual in base 10).

    i was wondering if some one could please give me a hint as how to start off or give me a push in the right direction as i don't have any idea how to start(go about the question).
    However, i do know that i need to show that 3|s and if this is true then 3|n will hold true as 3|n <=> 3|s.

    plus i know that

    n = ( a_k) * ( 10^k) + (a_(k-1)) * (10^(k-1)) + ... + a * 10 + ( a_0) * 10^0

    and

    s= ( a_k) + (a_(k-1)) + ... + a + ( a_0)

    oh and i was thinking that i may need to use the definition of divisibility where x|y <=> y = xn as

    3 | ( a_k) + (a_(k-1)) + ... + a + ( a_0) <=> ( a_k) + (a_(k-1)) + ... + a + ( a_0) = 3 m
    but then once again im lost in what to do


    thankyou very much,
    from an extremely stressed CoCo_RoAcH
    Last edited by CoCo_RoAcH; March 31st 2009 at 04:26 AM. Reason: forgot to include another thought ><
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by CoCo_RoAcH View Post
    hey! the question is as follows:

    Show that 3|n if and only if 3|s, where s is the sum of the digits of n (expressed as usual in base 10).

    i was wondering if some one could please give me a hint as how to start off or give me a push in the right direction as i don't have any idea how to start(go about the question).
    However, i do know that i need to show that 3|s and if this is true then 3|n will hold true as 3|n <=> 3|s.

    plus i know that

    n = ( a_k) * ( 10^k) + (a_(k-1)) * (10^(k-1)) + ... + a * 10 + ( a_0) * 10^0

    and

    s= ( a_k) + (a_(k-1)) + ... + a + ( a_0)

    oh and i was thinking that i may need to use the definition of divisibility where x|y <=> y = xn as

    3 | ( a_k) + (a_(k-1)) + ... + a + ( a_0) <=> ( a_k) + (a_(k-1)) + ... + a + ( a_0) = 3 m
    but then once again im lost in what to do


    thankyou very much,
    from an extremely stressed CoCo_RoAcH
    So:

    n \text{ mod } 3=\sum (a_i \text{ mod } 3) (10^i \text{ mod } 3)=\sum (a_i \text{ mod } 3)=\left(\sum a_i\right) \text{ mod } 3

    CB
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    So:

    n \text{ mod } 3=\sum (a_i \text{ mod } 3) (10^i \text{ mod } 3)=\sum (a_i \text{ mod } 3)=\left(\sum a_i\right) \text{ mod } 3

    CB
    Hey CaptainBlack
    thankyou very much for that formula but i won't be able to use it as we haven't learnt it.

    is there another way to start this question?

    thankyou, CoCo_RoAcH
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