1. Let m₁, m₂ € N and suppose that a, b = Z satisfy a = b (mod m₁) and a = b (mod m₂). Prove that, if ged(m₁, m₂) = 1, then a = b (mod m₁m₂). (Hint: translate the congruences to divisibilities and use Euclid's lemma.)

Q: 3: Suppose gEF is a generator and j,d€Z. (a) Please prove that gid=1 mod p if and only if jd=0 mod p…

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Q: The statement a ≡ b (mod n) is called a congruence.

A: Given that  The statement a ≡ b (mod n) is called a congruence.

Q: 3. Find integers x, y € [0,51] such that (a) 23x= 1(mod52) (b) 23y = 4(mod52)

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Q: Let a, b be integers. What does it mean for a = b (mod 17)? Describe the elements of Z17. Compute…

A: a ≡ b (mod 17) means that a − b = 17k for some k ∈ ℤ. ∴  b − a = −17k = 17(−k); henceb ≡ a (mod…

Q: ​Q3// Use Chinese theorem to solve the following congruence: X ≡ 2 mod 3 X ≡ 1 mod 5 X ≡ 2 mod 4

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Q: Find all x (mod 1999) such that x³ = 2 (mod 1999).

A: We use primitive roots to solve this problem.

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: Calculate C(12,5) by hand. As C(n,r) = n!/(n-r)!r! identify n and r, and write out every number in…

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Q: 1. Show that k1601 = k (mod 17) for any k.

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Q: Consider the linear congruence ax = b (mod m). If (a,m) = 1, then this has a unique solution x = a¹b…

A: Since gcd(a,m)=1. ad+mk=1 for some integers d and k. ⇒ad=1-mk⇒ad=1 (mod m) So a is invertible (mod…

Q: 2r = 3 4x = 2 3x 2 (mod 5) (mod 6) (mod 7).

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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: Let a and b be positive integers and let b= qa +r, where 0 ≤r <a. Prove that (a) 2b - 1 = 2" - 1…

A: Let  and  be two positive integers.Let  where (a) We  have to prove that      Now, observe that…

Q: For which integers c, 0<c < 30, does the congruence 12x == c (mod 30) have solutions? When there are…

A: Given congruence is 12x≡c(mod 30). We know that ax ≡b(mod m) have solutions when gcd(a,m) divides b.…

Q: 1. Prove that if ā and be Zn are invertible modulo n, so is ā - b. Is this the case for ā + b?…

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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: Furthermore, any two solutions x and y are congruent modulo n¡n2 · Solve the following systems of…

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

Q: The linear congruence ax = b(mod m) has a unique solutionif and only if: gcd (a,m) =1 gcd(a,m) =1 o…

A: First option is correct. gcd(a, m)=1.

Q: Prove that if gcd(a, m)=1 and gcd(a - 1, m) =1, then 1+a+a²+...+ a² + a(m)−1 = 0 (mod m).

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Q: Solve the congruence 11x = 45 (mod 97) where x is an integer in the interval [0,97).

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Q: Which of the following linear congruence has no solution? (a) 3x = 9 (mod21) (b) 6x = 4 (mod20) (c)…

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Q: 2. Let a be a positive integer. Explain why the linear congruence az b (mod a + 1) has a solution in…

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Q: Prove that pork = pur in O₂ if and only if 0 w = 2m for some m€ Z and k = 1 (mod 2)

A: To prove that ρθrk=ρωrl in Ο2 if and only if θ-ω=2πm for some m∈ℤ  and k¯≡ l¯mod 2.

Q: Which of the following linear congruence has 2 incongruent solutions? (a) 3x = 9 (mod21) (b) 14x =…

A: to solve linear congurence , ax≡b(mod m) .........(1) the first stem is to reduce the congurence  by…

Q: Prove that the number of solutions to the congruence xk ≡ 1 mod p is gcd(k, p − 1)

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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: 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: 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: The linear congruence ax = b(mod m) has a unique solutionif and onlyif: @cd(a,b) = 1 cd(a,m) =1…

A: Second option is correct.

Q: Let n be a positive integer and let ged(b, n) = 1. Show that bx by (mod n) if and only if x=y(mod…

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

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Q: 7.27. Theorem. Let p be a prime and let i and j be natural numbers with i ‡ j satisfying 1<i,j<.…

A: Suppose, i2≡j2 (mod p)⇒i2-j2 ≡0 (mod p)⇒(i+j)(i-j)≡0 (mod p)⇒p|i+j or p|i-j⇒p≤i+j or p≤i-j

Q: Let a , b >= 1 and x ∈ Z. Prove that if xb ≡ 1 (mod a) then gcd(x, a) = 1

A: Let a , b ≥ and x ∈ Z  Suppose xb ≡ 1 (mod a)  ⇒a divides xb-1  By defination of divisibility, there…

Q: for a fixed n>1 Show that all the solvable congruences x2 ≡ a ( mod n ) have the same number of…

A: To Proof: For a fixed n&gt;1 , all the solvable congruences x2 ≡ a ( mod n ) have the same number of…

Q: a.) Determine if x< = 2 (mod 7817) has solutions. b.) Determine if x< = 3760 (mod 4861) has…

A: x2≡2(mod 7817) has a solution if and only if  2 is a quadratic residue of 7817. Euler's criterion…

Q: 3. Exercise 52.3 #40. Prove that the system of linear congruences in one variable given by z= b, mod…

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Q: 2. Find all solutions of the linear congruence 12x = 9 (mod 15).

A: P

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: For which positive integers a is the congru

A: Given,               ax4≡2 (mod 13)