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  1. #1
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    plz help to solve

    find the remainder when 1!+2!+3!+..........99!+100! divided by 12.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by mythri sharma View Post
    find the remainder when 1!+2!+3!+..........99!+100! divided by 12.
    I am not positive...but would you apply Wilson's Theorem?
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  3. #3
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    Quote Originally Posted by mythri sharma View Post
    find the remainder when 1!+2!+3!+..........99!+100! divided by 12.
    Hint: n! is evenly divisible by 12 for all n > 3.
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    Quote Originally Posted by mythri sharma View Post
    find the remainder when 1!+2!+3!+..........99!+100! divided by 12.
    Try writing the factorials as a product.

    Now 12 = 4 x 3

    1! = 1
    2! = 2 x 1
    3! = 3 x 2 x 1
    4! = (4 x 3) x 2 x 1 <------ There is a 4 x 3 here
    5! = 5 x (4 x 3) x 2 x 1 <------ There is a 4 x 3 here
    6! = 6 x 5 x (4 x 3) x 2 x 1 <------ There is a 4 x 3 here
    7! = 7 x 6 x 5 x (4 x 3) x 2 x 1 <------ There is a 4 x 3 here

    Well I am sure you can see by now that 12 will divide any factorial greater than or equal to 4. That means the only fellows who can leave remainders are 1!+2!+3! = 1+2+6 = 9. So 9 is the remainder when divided by 12.

    Now can you try a related question:
    "Which is the largest factorial number that is not divisible by 27?"

    Try it
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Isomorphism View Post
    Try writing the factorials as a product.

    Now 12 = 4 x 3

    1! = 1
    2! = 2 x 1
    3! = 3 x 2 x 1
    4! = (4 x 3) x 2 x 1 <------ There is a 4 x 3 here
    5! = 5 x (4 x 3) x 2 x 1 <------ There is a 4 x 3 here
    6! = 6 x 5 x (4 x 3) x 2 x 1 <------ There is a 4 x 3 here
    7! = 7 x 6 x 5 x (4 x 3) x 2 x 1 <------ There is a 4 x 3 here

    Well I am sure you can see by now that 12 will divide any factorial greater than or equal to 4. That means the only fellows who can leave remainders are 1!+2!+3! = 1+2+6 = 9. So 9 is the remainder when divided by 12.

    Now can you try a related question:
    "Which is the largest factorial number that is not divisible by 27?"

    Try it
    Would it be 8! because since the only numbers(that you can't have a repeated factor) that multiply to 27 are 9*3 which only occurs first in 9!?
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  6. #6
    Lord of certain Rings
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    Quote Originally Posted by Mathstud28 View Post
    Would it be 8! because since the only numbers(that you can't have a repeated factor) that multiply to 27 are 9*3 which only occurs first in 9!?
    Yes, right
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  7. #7
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Isomorphism View Post
    Yes, right
    Hooray!
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