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Math Help - Prime numbers problem

  1. #1
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    Prime numbers problem

    If anyone could explain how this is done, it would be very much appreciated!

    Fermatís Little Theorem states that if a is an integer with gcd(a,p) = 1, where p is a prime number, then:
    a^(p-1) ≡ 1 mod p

    Define e_p(a) to be the smallest number n>0 such that a^n ≡ 1 mod p. Compute e_p(a) for the prime p = 5 and 1≤a≤p-1
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    Quote Originally Posted by clockingly View Post
    If anyone could explain how this is done, it would be very much appreciated!

    Fermatís Little Theorem states that if a is an integer with gcd(a,p) = 1, where p is a prime number, then:
    a^(p-1) ≡ 1 mod p

    Define e_p(a) to be the smallest number n>0 such that a^n ≡ 1 mod p. Compute e_p(a) for the prime p = 5 and 1≤a≤p-1
    There are only a limited number of possibilities here.

    Obviously 1 is a rather trivial example of this since 1^n = 1 for any n.

    We can look at 2 and get that 2^4 \equiv 1 \text{ mod 5}

    etc.
    3^4 \equiv 1 \text{ mod 5}
    and
    4^2 \equiv 1 \text{ mod 5}

    So I guess we have that
    e_p(1) = 1
    e_p(2) = 4
    e_p(3) = 4
    e_p(4) = 2

    Is this what you were looking for?

    -Dan
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