The number of number theory congruence type

The same remainder of the first section

theorem 1: If ac = BC (mod m), then A = B (mod m/(M, c))

section II remaining classes and total remaining systems

The total remaining system is the arbitrary number of each remaining class.

theorem 1: X over M of the complete residual system, then if there is an integer with M-A,ax is also over m of the complete residual system.

theorem 2: M1 m2 is a positive mode of two, then if X1 x2 respectively over M1 m2 of the complete residual system, M1X2 + m2x1 over m1m2 of the complete residual system.

the third section of the system and Euler functions

definition : First, for a modulo m the number of each remaining class is the same as M's mutual primality, and for each of the remaining classes that have an inter-primality with M, one is given a contraction of M.

definition : The Euler function Phi (n) represents the number of 1~n with N.

theorem 1: The number of Phi (m) in the indent of M

theorem 2: a[1] a[2] ... a[phi (m)] constitutes the contraction of M, equivalent to:

1. A[i] and M-biotin
2. A[i] and a[j] different, I <> j

theorem 3: x is the contraction of the M, then if there is an integer with M-A,ax also the contraction of M.

theorem 4: M1 m2 is a positive mode of two, then if X1 and X2 respectively over M1 and m2 of the contraction system, m1x2 + m2x1 over m1*m2.

The theorem of this section 3 4 is exactly similar to the second section Theorem 1 2.

theorem 5: "Euler's theorem": if (A, m) = 1, then POW (A, phi (m)) = 1 (mod m)

Proof: (a x1) (a x2) ... (a XP) = X1 x2 ... xp (mod m)

Pow (A, p) x1 x2 ... xp = x1 x2 ... xp

$ $a ^p = 1 (\mod m) $$

theorem 6: Phi (n) = Phi (p1 ^ a1 p2 ^ A2 ... pk ^ ak) = N (1-1/p1) (1-1/P2) ... (1-1/PK)

With the support of Theorem 3 4, theorem 5 6 becomes apparent.

The number of number theory congruence type