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4 a = 11, n = 30, we have 11φ(30) ≡30 118 ≡30 1214 ≡30 14 ≡30 1 As a preclude to launching our proof of Euler s Generalization of Fermet s theorem , we require a preliminary lemma Lemma Let n > 1, gcd(a,n)=1, if m1 , m2 , . . , mφ(n) are the postive integers less than n and m1 , m2 , relatively prime to n , then am1 , am2 , am3 , . . , amφ(n) are congruent modulo n to . . , mφ(n) in some order. if gcd (a, n) = 1 , and Let Φ(n) = { m1 , m2 , . . , mφ(n) } Then {ami | mi ∈ Φ(n)} ≡n Φ(n) in some order Proof f act1 Observe that no two of the integers am1 , ami ≡n amj for all otherwise mi ≡n mj gcd(mi , n) f act2 since gcd (a, n) = 1 φ(n) , from these two facts ami ≡n mj ∈ Φ(n) This proves that the number am1 , am2 , am3 mφ(n) are identical ( modulo n ) in certain order.

1, this LDE can be solved iff gcd(a, m) = 1. 4 If p is prime, then all elements in Zp except 0 have multiplicative inverses. Note that by Property 1, it is clear that Zm , +, 0 and Zp − {0}, ∗, 1 (where p is prime) are abelian groups. Further, Zp , +, ∗, 0, 1 is a commutative ring. We now come to solving single variable linear congruences and demonstrate the correspondence between the congruences and LDEs. 5 ax ≡m b has a solution iff gcd(a, m)|b. If d = gcd(a, m) and d|b then ax ≡m b has d mutually incongruent solutions modulo m.

Note that by Property 1, it is clear that Zm , +, 0 and Zp − {0}, ∗, 1 (where p is prime) are abelian groups. Further, Zp , +, ∗, 0, 1 is a commutative ring. We now come to solving single variable linear congruences and demonstrate the correspondence between the congruences and LDEs. 5 ax ≡m b has a solution iff gcd(a, m)|b. If d = gcd(a, m) and d|b then ax ≡m b has d mutually incongruent solutions modulo m. 1. 2, we know that all solutions of this LDE are given by: xu = x0 + (m/d)u, yu = y0 + (a/d)u.