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Math Help - Final Digit of 3^1001

  1. #1
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    Final Digit of 3^1001

    Explaining your logic, what is the final digit of the number 3^1001?


    *I'm guessing that they don't want you to use a calculator?



    Thank you.
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  2. #2
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    Hello,

    Quote Originally Posted by gearshifter View Post
    Explaining your logic, what is the final digit of the number 3^1001?


    *I'm guessing that they don't want you to use a calculator?



    Thank you.
    You can see it by iterations...

    3^1=\bf{3}
    3^2=\bf{9}
    3^3=2\bf{7}
    3^4=8\bf{1}
    3^5=24\bf{3}
    3^6=72\bf{9}

    So you can see that there are only 4 possible values : 1;3;7;9. When you multiply by 3 a number which finishes by 3, the result will finish by 9. When you multiply by 3 a number which finishes by 9, the result will finish by 7, etc... You can prove it by writing that these numbers are 10k+3 for example.. When multiplying by 3, it yields 30k+9, which obviously has its final digit number equal to 9
    You can notice that the same value comes back every time you add 4 to the power.

    For example : 3^1, 3^5, 3^9, \dots will have the same final digit number.

    So continuing this way, if I add 25 times 4 to the power, it will be 3^{101}, which will have the same final digit number as 3^1

    What is it going to be for 3^{1001} ?



    (I hope this is clear enough :/)
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