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Math Help - Units digit of m^5 - m is 0

  1. #1
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    Units digit of m^5 - m is 0

    Dear All,

    A question that I have been trying to prove ;

    prove that all postive numbers (m), the units digit of m^5 - m is 0 ?

    Since the integer is in base 10,
    suppose m=0 then (0)^5 - 0 = 0 ( this is not a postive int)
    m=1 then (1)^5 - 1 = 0
    m=2 then (2)^5 - 2 = 30 ....
    .......

    using (10m + u)^2 is a method of proving however I have yet to prove it for this question.

    regards

    S.J
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  2. #2
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    Quote Originally Posted by srvsrv2 View Post
    A question that I have been trying to prove ;
    prove that all postive numbers (m), the units digit of m^5 - m is 0 ?
    That is equivalent to showing m^5-m is a multiple of ten.
    Use induction.
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  3. #3
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    Quote Originally Posted by srvsrv2 View Post
    Dear All,

    A question that I have been trying to prove ;

    prove that all postive numbers (m), the units digit of m^5 - m is 0 ?

    Since the integer is in base 10,
    suppose m=0 then (0)^5 - 0 = 0 ( this is not a postive int)
    m=1 then (1)^5 - 1 = 0
    m=2 then (2)^5 - 2 = 30 ....
    .......

    using (10m + u)^2 is a method of proving however I have yet to prove it for this question.

    regards

    S.J


    m^5-m=m(m-1)(m+1)(m^2+1) . It's easy now to check that this number is even and a multiple of 5 for any integer value of m.

    Tonio
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  4. #4
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    You can prove it using Fermat's Little Theorem or By mathematical induction, both works.
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  5. #5
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    Thanks. I shall be posting my results very shortly.
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  6. #6
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    Quote Originally Posted by srvsrv2 View Post
    Thanks. I shall be posting my results very shortly.
    you can try it by recalling that m^5 and m have the same unit digits (this is a nine-timed repetition argument)
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  7. #7
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by teachermath View Post
    you can try it by recalling that m^5 and m have the same unit digits (this is a nine-timed repetition argument)
    Assuming you mean to check each of the residue classes this is technically a "ten-timed" repetition argument.
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