For any a, b∈Z and positive integer n, we say that a is congruent to b module N, written a≡b (mod n), if, and only if, (b−a) mod n=0 (i.e. n|(b−a) ). Prove that for any a, b, c, d∈Z, if a≡c (mod n) and b≡d (mod n), then a+b≡c+d (mod n).

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: Given |a,b,c,n €Z, show that, if ac bc (mod n), a = b (mod), d= mdc(c, n). then

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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: Problem MCD = Greatest common divisor (GCD) t them.a, b, c and m €Zsuch that m 22. a-). Prove that a…

A:

Q: Prove for any integers a and b, if (a3+b3) mod 3 is not 0, then (a+b) mod 3 is not 0. (Hint: recall…

A: Congruence Modulo Relation on Integers: Let a, b∈ℤ and let m be a fixed positive integer.  Then a…

Q: If a cy (mod n) and c> 0 then a) x y (mod en) b) x y (mod n) ) x=y (mod) d) None of these :Select…

A:

Q: 4. For any integer n, show that if n³ = 1 (mod 47), then n = 1 (mod 47).

A: To show that if n3≡1(mod47), then n≡1(mod47), it can use proof by contradiction.Assume that…

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: 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: consider the statement: for all n∈N , x∈Z , x^2 ≡ 1 mod n (        ) x ≡ 1 mod n or x≡ -1 mod n a)…

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Q: 1.Let a, b, and c be integers, where a≠0. If a | b and a | c, then a | (b + c)            True or…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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).…

A:

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 a, b e Z and n E Z*. Prove that if a = b(mod n), then gcd(a, n) = gcd(b, n). (Hint: use the…

A: The greatest common divisor (gcd) of two or more numbers is the greatest common factor number that…

Q: Suppose x F and you discover that x¹ = 1 mod p and x² = 1 mod p. Please prove that xgcd(m,n) = 1 mod…

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Q: Let a, b, c, m be integers with m > 2. Prove that if ac ≡ bc mod m and d = gcd(m, c) , then a ≡ b…

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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: Let a, b e Z and n E Z*. Prove that if a = b(mod n), then gcd(a, n) = gcd(b, n).

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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…

A:

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…

A:

Q: Let 2,y EZ and nEN. Find which is true ot false. Show your answer with proof or counter example (if…

A: a) It is true Given x≡y(mod n)So, we've x = y + nk for some k∈ℤConsider gcd(x,n) = a , and gcd(y,n)…

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

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Q: Suppose that each of a, b, and c is an integer and m is a positive integer. Show that if a = b(mod…

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

A:

Q: true or false  1..The GCD of -50, -100 and -125 is -25 2.If (a, b) = d and d│m, then (ax is…

A: Solution

Q: If gcd(m,n) = 1 is a given condition, how can you prove that the congruences x ≡ a (mod m) and x ≡ b…

A: The congruences have a solution when .

Q: Let n1, n2, n3 be three numbers that are pairwise relatively prime so that (n1, n2) = (n2, n3) =…

A: Given that n1,n2 and n3 are three numbers that are pairwise relatively prime. We have to prove…

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: 20. If a € Z and a = 1 (mod 5), then a² = 1 (mod 5). 23. Let a, b, c € Z and n EN. If a = b (mod n),…

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Q: a₁xb₁ (mod m₁) a2xb₂ (mod m₂) ax=bk (mod mk.) has a unique solution modulo M = [1 mi, provided that…

A: This is called as the Chinese Remainder Theorem

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.