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Topic: I need to demonstrate that 3^80 -2^20 is divisible by 5 . hint u(3^80)=1
Replies: 1   Last Post: Oct 5, 2012 4:13 PM

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Piotr

Posts: 2
Registered: 9/19/12
Re: I need to demonstrate that 3^80 -2^20 is divisible by 5 . hint u(3^80)=1
Posted: Oct 5, 2012 4:13 PM
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3^80 -2^20 is divisible by 5 if

3^80 = 2^20 (mod 5)

now observe that:

k - natural

(3^i) mod 5 = [nb you can call this function u]
1 for i=4k
3 for i=4k+1
4 for i=4k+2
2 for i=4k+3

and

(2^i) mod 5 =

1 for i=4k
2 for i=4k+1
4 for i=4k+2
3 for i=4k+3


so


(3^80) mod 5 = (3^(4*20+0)) mod 5 = 1
(2^20) mod 5 = (2^(4*5+0)) mod 5 = 1

and we are done



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