A number theory problem by Dennis Escobal Sabanto
Number Theory
Level
pending
25 TO the Power OF 2015 mod 18
Find The Remainder.
15
23
33
13
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1 solution
GCD(25,18)=1
Thus we can use Euler here to simplfiy
Totient(18)=6 thus
25^6 = 1 mod 18
2015 = 335*6+5 thus
25^2015 = 25^(335 6+5) = (25^5) 25^(335*6)=25^5 mod 18
25=7 mod 18
Thus we have
25^2015 = 7^5 mod 18
7^2 = 49 mod 18 = 13 mod 18
7^5 = 7 7^2 7^2=7 13 13 mod 18= 13 mod 18
Thus 25^2015 = 13 mod 18