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Math Help - Find the last digit

  1. #1
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    Question Find the last digit

    Find the last digit of 541^(341). Clearly, the last digit would be one, but I don't know how to go about showing that it would be one. Can someone help please?
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  2. #2
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     541^{341} \equiv 1^{341} \equiv 1 \bmod{10} i.e. the last digit is going to be 1. If this doesn't make sense to you prove it by induction. I.e. prove that 541^n  has ending digit one for all n.
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  3. #3
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    by induction : 541^n ,
    for n = 1 , 541^1 = 541 => it ends with 1 true
    assume that 541^k ends with one is true and proof that 541^(k+1) ends with one
    now 541^(k+1) = 541^k * 541 ,,, we prooved that 541^k ends with one and 541 ends with one the multiplication of two numbers that ends with one will result in a number that ends with one .
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