By Florentin Smarandache

Articles, notes, generalizations, paradoxes, miscellaneous in arithmetic, linguistics, and schooling.

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An' xn ≡ b ' (mod m) , which is a linear equation; when it is resolved in Z it has the general solution: ' i 62 ⎧ x1 = α11k1 + ... + α1n kn + γ 1 ⎪ ⎪⎪⋅ ⎨⋅ ⎪ x = α k + ... + α k + γ n1 1 nn n n ⎪ n ⎪⎩ y = α n +1,1k1 + ... + α n +1,n kn + γ n+1 k j being parameters ∈Z , j = 1, n , α ij , γ i ∈ Z , constants, i = 1, n + 1 , j = 1, n . Let’s consider α ij' ≡ α ij (mod m) and γ i' ≡ γ i (mod m) with 0 ≤ α ij' , γ ' ≤ m − 1; i = 1, n + 1, j = 1, n . Therefore ⎧ x1 = α 11'k1 + ... + α1n' kn + γ 1' (mod m) ⎪ ⎪⋅ ; k j = parameters ∈Z, j = 1, n ; (**) ⎨ ⎪⋅ ' ' ⎪ xn = α n1 k1 + ...

3. If E1 (n) and E2 (n) belong to ε (n) with d 0 E1 (n ) = r1 and d 0 E2 (n ) = r2 , c( E1 (n )) = a1 and c( E2 (n )) = a2 , then: a) E1 (n )E2 (n ) ∈ ε (n ) ; d 0 ( E1 (n )E2 (n )) = r1 + r2 ; c( E1 (n )E2 (n )) which is a1a2 . E (n) b) If E2 (n ) ≠ 0 ∀n ∈ N(n ≥ nE2 ) , then 1 ∈ ε (n) and E2 (n) ⎛ E (n) ⎞ ⎛ E (n) ⎞ a d 0 ⎜ 1 ⎟ = r1 − r2 , c ⎜ 1 ⎟ = 1 . ⎝ E2 (n) ⎠ ⎝ E2 (n) ⎠ a2 c) If α is a real constant and if the operation used has a sense (E1 (n))α (for all n ∈ , n ≥ nE1 ), then: ( E1 (n) ) α ∈ ε ( n) , d 0 ( ( E ( n) ) ) = r α , c ( ( E ( n) ) ) = a α 1 α 1 1 α 1 d) If r1 ≠ r2 , then E1 (n) ± E2 (n) ∈ ε (n) , d 0 (E1 (n ) ± E2 (n )) is the max of r1 and r2 , and c (E1 (n ) ± E2 (n )) = a1 , respectively a2 resulting that the grade is r1 and r2 .

D. Note 2. To prove the following theorem, we suppose known the criterion of convergence of the series and certain of its properties (3) Theorem of convergence and applications. Theorem: Let’s consider E ( n) ∈ ε ( n) with d 0 (E (n )) = r having the series ∑ E(n) , E(n) ≡/ 0 . n ≥ nε Then: A) If r < −1 the series is absolutely convergent. B) If r ≥ −1 it is divergent where E(n) has a sense ∀n ≥ nE , n ∈ N . Proof: According to lemmas 1 and 2, and because: the series ∑ E (n) converge ⇔ the series − ∑ E (n) converge, n ≥ nE we can consider the series n ≥ nE ∑ E (n) like a series with positive terms.