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Math Help - Divisible or Remainder??

  1. #1
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    Divisible or Remainder??

    Does 9 divide 5^(23) + 1? If not, what is the remainder?


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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Note  23 = 16+4+2+1 (all powers of  2 ). So  5^{23} = 5^{16}\cdot 5^4\cdot 5^2\cdot 5^1 .

     5^1 \equiv 5 \mod{9}
     5^2 \equiv 7 \mod{9}
     5^4 =\left(5^2\right)^2 \equiv 7^2 \equiv 4 \mod{9}
     5^8 =\left(5^4\right)^2 \equiv 4^2 \equiv 7 \mod{9}
     5^{16} =\left(5^8\right)^2\equiv 7^2 \equiv 4 \mod{9}

    Thus  5^{23}+1 \equiv 4\cdot 4\cdot 7\cdot 5+1 \equiv 3 \mod{9} .
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