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    Excuse me for being stupid but i'm totally new to this kind of problem and needing some help.. here's what I've done so far...

    2^55 + 1

    = (2 + 1) (2^54 - 2^53 + 2^52 - ... + 2^4 - 2^3 + 2^2 - 2^1 + 1)

    (2^4 - 2^3 + 2^2 - 2^1 + 1) = 11

    so 2^55 + 1

    = 3(2^54 - 2^53 + 2^52 - ... + 2^5) + 3(11)

    = 3(2^54 - 2^53 + 2^52 - ... + 2^5) + 33


    and now i'm stuck! and probably barking up the wrong tree.
    any help appreciated, thanks.
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    erm, i think this is how you do it....

    2^55= (2^5)^11=(32)^11

    therefore from fermat's little theorem (32)^11=32(mod 11) so adding one gives 33(mod 11).... which is 0(mod 11)

    all odd powers of 2= 2(mod 3) so adding one gives 0 (mod 3)...

    this makes 2^55+1= 0(mod 11) + 0(mod 3) = 0 (mod 33)
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    Even if you've never met FLT it's ok as the numbers are so small.

    Mod 3 and Mod 11 both:-

    2^55 + 1 = (2^5)^11 + 1 = 32^11 + 1 = (-1)^11 + 1 = -1 + 1 = 0

    As 3 and 11 both divide 2^55+1 then 33 does.
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    In answer to a PM of Sebbie on modding numbers - in this specific case we write:-

    a = b mod 3 when a-b is a multiple of 3.

    It's fairly easy to check that if

    a = b mod 3

    the

    a^n = b^n mod 3 for any natural number n.

    So in our example 32 = -1 as the difference is 33 = 3 x 11.

    In mod 3 arithmetic there are just three types of number, those of the form

    3k, 3k+1, 3k+2.

    In mod 11 there are 11 types

    11k, 11k+1,... 11k+10.

    These are just generalisations of odd and even numbers (which make up mod 2 arithmetic).

    There's stuff on the web about this at:-

    http://www.maths.ox.ac.uk/prospectiv...n/lecture1.pdf
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    hey thanks a lot guys.. bit of a delayed reaction from me here but there we are!
 
 
 
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