What is the remainder when 17^(100000000…), with n zeroes is divided by 20? I.e.: 17^(10^n)mod20= ?

Haven Kerr 2022-09-24 Answered
What is the remainder when 17 100000000 , with n zeroes is divided by 20? I.e.:
17 10 n mod 20 =   ?
My approach: I can solve it by Euler's totient function ; 17 8 = 1 mod 20, so remainder is 1.
But I want to know if the following method will fetch the result: 17 divided by 20 gives −3 as remainder and −3 raised to any multiple of 10 mod 20 gives 1?
Is this logic correct? I think it works only for positive remainder.
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Answers (2)

Simon Zhang
Answered 2022-09-25 Author has 7 answers
Modular arithmetic doesn't care about positive or negative. 17 and −3 behave in exactly the same way modulo 20. They are both representatives of the modulo 20 conjugacy class
{ , 23 , 3 , 17 , 37 , }
and thus one might even say they are equal, under some interpretations of modular arithmetic (they are equal "in Z 20 ", for those who know what that means). So yes, that would work.
Note that ( 3 ) 2 = 9 = 10 1, so
( 3 ) 4 = ( 10 1 ) 2 = 100 20 + 1 1 ( mod 20 )
is probably the fastest way to get to your answer.
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maredilunavy
Answered 2022-09-26 Author has 2 answers
Hint   gcd ( a , 20 ) = 1 x 4 1 ( mod 20 ) , being true mod 4 and 5.
Thus if f n = a 10 n then mod 20 :   f n + 1 f n 2   by f n + 1 = f n 10 ( f n 2 ) 4 f n 2 f n 2
So n 0 1 2 3 f n a a 2 1 1 by s q u a r i n g repeatedly.
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