4. Prove that if a = b (modn) and m|n, then a = b (modm)…....

Q: e) Define W: Z -Z and V: Z → Z by the rules W(a) = a² and V(a) = a mod 5 for all integers a. Find…

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Q: 6. Let m, n EN with m, n > 2, and let a, b E Z. Prove that a = b (mod mn) implies a = b (mod m) and…

A: (a) a = b (mod mn)       => there exists some integer k such that                    a = b +…

Q: Let x E Z and y E Z. Prove the following statement: If y = 2 mod 4, then xy 1 mod 4.

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Q: Za) Let A = {n E Z | n = 0 (mod 2)} and B = {n € Z | n = 0 (mod 4)}. Prove that if n E (A – B), then…

A: Follow the steps below.

Q: Let a, be Z and n E N. For each statement decide whether it is necessarily true, or whether it can…

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Q: Question 2 Let a, me Zt and (a,m) = (a-1,m) = 1. Then prove that 1+a+...+am)! = 0(modm) E

A: Eulers theorem will be used to solve the problem.

Q: Let a, b, n be positive integers. Prove that if a = b (mod n), then GCD(a, n) = GCD(b,n).

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Q: 1. Let a, b, c, r, s, and t be integers with r>s>t>1. If a = b (mod r), a = c (mod s), and gcd(r,s)…

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Q: Find n ∈ {1, 2, 3, . . . , 28} that satisfies 26! ≡ n (mod 29).

A: Given ,              26! ≡ n mod(29)  Then we have to find the  value of n∈  1 , 2 , 3 , .........,…

Q: (b) Prove that if a = b (mod m) and if b = c (mod m), then a = c (mod m).

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Q: (c) For each integer a, if a 4 (mod 8), then a 2 (mod 8).

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Q: Let A = {r € Z | r mod 15 = 5} and B = {x € Z| r mod 3 = 2}. Give an outline of a proof that A C B,…

A: Suppose we have two non-empty sets A and B . If x∈A then x∈B for all x∈A then A⊆B . If x∈B then x∈A…

Q: 1. Let a, b e Z. Prove that if a = 0 (mod 5) and b = 2 (mod 5), then a² + b² = 4 (mod 5).

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Q: 4. Let a, b, c, d, n € Z such that a = b (mod n) and c= d (mod n). (a) Prove that a+c=b+d (mod n).…

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Q: Q6. Using the definition of x = y mod N, justify the rule x = x' mod N, y = y' mod N = x x x' = y x…

A: Using the definition of modulo N.

Q: Let a, b, c and meZ, and m 2 2 1. Prove: if a =b (mod m), then GCF (a, m) = GCF (b, m) %3D m 2.…

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Q: 5. Let n be a positive integer, and assume that a and b are integers such that a = b (mod n). Prove…

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Q: Q3 Not submitted Section 4.1 # 14 Show that if a, b, and e are integers with e > 0 such that a = b…

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Q: Prove that if a = b (mod m) and if c = d (mod m), then a+c=b+d (mod m).

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Q: 4 If a = b (mod n), then a" = b™ (mod n) for every positive integer m.

A: It is given that  a≡b (mod n), then am≡bm (mod n) for every positive integer m. It will prove by…

Q: 2. Prove that if ar b (mod n), then () x = () (mod ()) where d = gcd(a, n). Apply this result to 5x…

A: We first find the gcd of the given numbers and then use that to solve the given congruence. The…

Q: (12) Let a,b, m, nE Z. Assune if greater than ave Prove mln and a = b (mod n) that then a = b (mod…

A: Congruence equations

Q: 2. Prove that if a = b( mod r) and c = d( mod r) then a – c= b – d( mod r)

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Q: Prove that if a = b (mod m) and if c = d (mod m), then ac = bd (mod m).

A: a≡b   (mod m)⇒ m|(a−b)That is there exist an integer k such that a−b=mk

Q: How many positive integers n less than or equal to 11,347 satisfy d(n) 1 (mod 2)?

A: We need to find a number of positive integers n≤11347 such that d(n)≡1(mod 2) i.e., , k∈Z.So, we…

Q: 2. Let A = {n? +n € Z:n e Z} and B = {r e Z : r³ – 3x³ + 2x = %3D O mod 5}. Prove or disprove: A C…

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Q: 7. Let n E Z. Prove that 9n3 6 (mod 27).

A: The objective is to prove the above statement.

Q: Let c, d e Z and m e N. For each statement decide whether it is necessarily true, or whether it can…

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Q: 3. For the following statements, either prove that they are true or provide a counterexample: (a)…

A: Given that, Let a, b, m, n ∈ Z such that m, n > 1 and n | m. a ≡ b (mod m) By using the…

Q: VI. Determine four values of x that will satisfy the following. 16. x = 4 mod 5 & x = 3 mod 7

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Q: Suppose gcd(a, n) = 1. Then a(") = 1 (n

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Q: 1. Let m be a positive integer. If a = b (mod m) and c= d (mod m), a) Prove a+c= b + d (mod m) b)…

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Q: Let x, y, u, v € Z and d E N. Prove that if x =y (mod d) and u = v(mod d), then xu = yv (mod d).

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Q: If nez then I89?bmodm a=+b moad m

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Q: Ifx= y (mod n) and u =v (mod n) then which of the following is false. (a) x+u= y+ v (mod n) (b) хи в…

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Q: 12.55. Prove the following: Let a, b, m, n e Z, where m, n 2 2. If a = b (mod m) and a = b (mod n),…

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