Let a,m be integers with m>0, gcd(a,m)=1 and gcd(a-1,m)=1. Prove that a^(phi(m)-1)+a^(phi(m)-2)+...+a^2+a+1 = 0 mod m

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- September 29th 2008, 08:48 AMmndi1105Congruences
Let a,m be integers with m>0, gcd(a,m)=1 and gcd(a-1,m)=1. Prove that a^(phi(m)-1)+a^(phi(m)-2)+...+a^2+a+1 = 0 mod m

- September 29th 2008, 10:49 AMwisterville
Hello,

(a^(phi(m)-1)+a^(phi(m)-2)+...+a^2+a+1)(a-1)=a^(phi(m))-1.

Now, divide by a-1 (which is prime relative to m.)

Bye.