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Math Help - divisibility

  1. #1
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    divisibility

    Find the last digit of the number 2^555666777
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  2. #2
    Super Member Bacterius's Avatar
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    2^1 \equiv 2 \pmod{10}
    2^2 \equiv 4 \pmod{10}
    2^3 \equiv 8 \pmod{10}
    2^4 \equiv 6 \pmod{10}
    2^5 \equiv 2 \pmod{10}
    2^6 \equiv 4 \pmod{10}
    2^7 \equiv 8 \pmod{10}
    2^8 \equiv 6 \pmod{10}

    Do you get the idea ?
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  3. #3
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    no i dont understand the mod 10 part...we have not learned that in class yet.

    so would my number end in 2 since the exponent is divisible by 9?
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  4. #4
    Super Member Bacterius's Avatar
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    Okay.

    You are looking for the last digit of 2^{555666777}. You don't need all the other digits. So, you can consider your number modulo 10 (which means you only consider the remainder of the number when divided by 10). For instance, 13 \equiv 3 \pmod {10} because 13 divided by 10 leaves a remainder of 3. And this gives us the last digit.

    Now what I've shown you in my preceding post basically tells us that given 2^a :

    - if a \equiv 0 \pmod{4} then the last digit of 2^a is 6.
    - if a \equiv 1 \pmod{4} then the last digit of 2^a is 2.
    - if a \equiv 2 \pmod{4} then the last digit of 2^a is 4.
    - if a \equiv 3 \pmod{4} then the last digit of 2^a is 8.

    And in your case, 555666777 \equiv 1 \pmod{4}, and therefore the last digit of 2^{555666777} is 2.

    Does it make sense ?
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  5. #5
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    so i need to take 555666777/10 and since my remainder is 1, the last digit of my number ends in a 2. yes i do understand now thank you so much.

    another question for clarification on my part. if i were taking 11^597637 and dividing it by 4, would the remainder of this always be 3, no matter what the exponent is?
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  6. #6
    Super Member Bacterius's Avatar
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    so i need to take 555666777/10 and since my remainder is 1
    No ! 555666777/4 gives a remainder of 1.

    if i were taking 11^597637 and dividing it by 4, would the remainder of this always be 3, no matter what the exponent is?
    The method I exposed here only works for some particular cases (here 2) because of special properties. But in the example you just gave above, using Euler's Theorem, given that \varphi{(4)} = 1, we can conclude that 11 raised to any power will give a remainder of 3 when divided by 4. But again, this is a little more advanced !

    I encourage you to learn modulus (congruences) and all associated theorems asap, it really helps a lot in number theory.
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  7. #7
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    okay thank you
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