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

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Q: et x, y E Z. We have (x+y)³ = (x+y)² (x+y) = (x² +2xy+y?)(x+y) = (x³+3x?y+3xy² + y³) = x³ + y3 od 3.

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Q: How do I prove this:  Let d ∈ N and let a, b ∈ Z. Prove or give a counterexample: If a 6≡ 0 (mod d)…

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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…

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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: 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: Suppose m, n e Z† are relatively prime. Prove that for all a, b e Z, a = b (mod mn) iff a = b (mod…

A: The congruence relation a≡b mod n implies that a-b is a multiple of n. That is n| a-b or a=kn+b If…

Q: For each statement, answer either true or false and include a brief/simple justification. We are NOT…

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Q: 15 Composite Then and O modn Show that if n there are with X#0 od n and y7 but where yy= Omod n.

A: Given n is a composite number

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: 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: Exercise 1. Suppose that a and b are integers, a = 3(mod11) and b = 8 (mod11) and c = a3 - b3. Then…

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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: 5. The Fibonacci numbers are defined as: fo = 0 , fi = 1 , and fn = fn-1 + fn-2 , for n > 2. Give a…

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Q: let a,b,c,d, and n be integers with n>1. Prove that if a is congruent to b(mod n) and c is congruent…

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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…

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Q: Exercise 5.2.14. Suppose January 25 is a Thursday. (a) Use Definition 5.2.6 to determine whether…

A: Two integer a and b are equivalent mod m of both a and b have the same remainder when divided by m.…

Q: 6. Prove that if gcd(a, b) 1, then gcd(a +b, ab) = 1. %3D %3D

A: Given that gcd(a,b)=1 If possible , let us assume that  gcd(a+b,ab)=d,d≠1 Then the number d must…

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

A: The objective is to prove the above statement.

Q: Exercise 5.2.4. Let n, q € N, and let a, b = Z. Suppose that a = b (mod n), and that q|n. Prove that…

A: Our objective is given below:

Q: If a =b (mod m) then a" = b" (modm) Vn e Z*.

A: Given a≡bmod m. we have to show that an≡bnmod m. we wil prove this by result by using method of…

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: Prove directly from the definition of congruence modulo n that if a,c, and n are integers,n >1, and…

A: Let a, c and n be integers such that n>1. We have to prove that if a≡cmod n, then a3≡c3mod n…

Q: aibic, and m are integers such that IF 9,b,c, and m m70, dc, d|m, and ac = bc (mod m), then a =b…

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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: Prove directly if the statements that are true, give counterexamples to disprove those that are…

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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…

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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: 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”.