[PDF] Congruence and Congruence Classes
Theorem 11 10 If a ≡ b (mod n) and c ≡ d (mod n), then (i) a + c ≡ b + d (mod n) (ii) ac ≡ bd (mod n) The congruence class of a modulo n, denoted [a], is the set of all integers that are congruent to a modulo n; i e , [a] = {z ∈ Z a − z = kn for some k ∈ Z}
[PDF] Math 371 Lecture §21 - BYU Math Department
metic of Z Theorem 2 2 If a ≡ b (mod n) and c ≡ d (mod n), then (1) a + c ≡ b + d (mod n), and (2) ac ≡ bd (mod n) Proof We suppose that a − b = nk for
[PDF] 3 Congruence
Theorem 3 3 If a ≡ b mod n then b = a + nq for some integer q, and conversely Similarly, multiplying, we get bd = (a + nq1)(c + nq2) = ac + naq2 + ncq1 + n 2 Now suppose that we wish to solve the congruence ax ≡ b mod n where d
[PDF] Congruences (Part 1) - Mathtorontoedu - University of Toronto
Eg 7 ≡ 3 (mod 4) -2 ≡ 6 (mod 4) 5 ≡ 9 (mod 4) 2) If a ≡ b (mod m) c ≡ d ( mod n) then ac ≡ bd (mod m) Proof: a = b + qm c = d + rm, some r,q ac – bd = (b
[PDF] MTHSC 412 Section 21 --Congruence and Congruence Classes
a is congruent to b modulo n and write a ≡ b (mod n) when Suppose that a ≡ b (mod n) and c ≡ d (mod n) Then a + c ≡ b + d (mod n) and ac ≡ bd (mod n)
[PDF] Number Theory
If a ≡ b (mod m) and c ≡ d (mod m), then ac ≡ bd (mod m) If a ≡ b (mod m), Theorem: An integer n is divisible by 11 i the di erence of the sums of the odd
[PDF] Modular Arithmetic - Cornell CS
12 nov 2014 · Let a, b ∈ ℤ, m ∈ ℕ a and b are said to be congruent modulo m, written a ≡ b ( mod m), if and only if a – b is If a ≡ b (mod m) and c ≡ d (mod m), then – a + c ≡ b + d (mod m) – ac ≡ bd (mod m) E g 11 ≡ 1 (mod 10) ⇒
[PDF] Congruences
integers a, b are congruent mod n, which is written as a ≡ b (mod n), if nb − a Example 2 For all a, b, c ∈ Z, if a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c ( mod n) multiplication by a, since if ab ≡ ac (mod n), then multiply by x to get x( ab) ≡ x(ac) ax ≡ b (mod n) has a solution if and only if d = gcd(a, n) divides b
[PDF] READING TO ACCOMPANY CSU-P MATH 319 - poritznet
17 fév 2014 · (9) If a ≡ b (mod n) and c ≡ d (mod n) then ac ≡ bd (mod n) Proof (1) Given a , b, c ∈ Z and n ∈ N, define d = gcd(a, n) ∈ N If ab ≡ ac
[PDF] if ac ≡ bc (mod m) and (c
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