Let A be a set of 6 integers. Prove that at least two elements of A belong to the same congruence class mod 5.

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: Let m be a positive integer. Show that a mod m = b mod m if a ≡ b (mod m)

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Q: For any integer a, prove that a = 0 or 1 (mod 3).

A: Every number is either of the forms 3k, 3k−1, 3k+1 where k is an integer

Q: The equivalence relation a = y(mod4) is defined on the set Z of integers. Determine the quotient…

A: The solution is below.

Q: If al = b (mod p), then prove that a = b (mod p).

A: (ii) given that                                                ap≡bp   (mod p) we have to prove…

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: Let x be an integer. Prove that, if mod3=2, then 3|(x2-1).

A: I hove used the theorem on congruent

Q: Let r be a primitive root of some n > 3. Prove that r = -1 (mod n).

A: Solution: Let n be any positive integer. Let Un= r : r is less than n and prime to n , where r=r+kn…

Q: a be an odd integer. Pro

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Q: Prove that for any integer k, at least one of k, k + 2, or k + 4 is divisible by 3. (Hint: consider…

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Q: For any integer a, prove that a=0 or 1 (mod 3).

A: The solution is given as follows

Q: Prove that ifn > 1 and a > 0 are integers, and d = GCD(a,n), then the additive order of a modulo m,…

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Q: . Show that Z with addition modulo m, where m≥ 2 is an integer, satisfies the closure, associative,…

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Q: Let A1, A2, . . . , An be sets. Prove that for every integer n ≥ 1, we have: (A1 ∪ A2 ∪ ··· ∪ An)c =…

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Q: Prove that if p is a prime and a is any integer so that p does not divide a, then the additive order…

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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 be a quadratic residue of the odd prime p. (1) Prove that a is not a primitive root modulo p…

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Q: 13. a. Prove that for every integer n 1, 10" = (-1)" (mod 11). |

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Q: Suppose r is a primitive root mod a prime p such that p = 3 mod 4. Prove ord,(-r) =

A: Given: p≡3mod 4 To find: ordpp-g=p-12

Q: Prove that for every prime p, there exists an integer r, such that r = 16 mod p.

A: To prove: For every prime p, there exists an integer x, such that x8≡16 modp.

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 m)…

A: Given: a,b∈Z and  m,n∈Z+ are relatively prime .

Q: Exercise 16. Prove (without induction) that B" = 0 mod B for all positive integers B and n. Hint:…

A: Congruence and proof without induction

Q: Show that if a,b, and n are integers such that n≥2 and a≡b (mod n), then gcd(a, n) = gcd(b, n).

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Q: Prove that Rn+3/Rn and R3 are isomorphic rings for all n belonging to N

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Q: 13. a. Prove that for every integer n > 1, 10" = (-1)" (mod 11).

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Q: 4. Exercise $2.5 #58a. Let a and b be integers not divisible by the prime mumber p. (a) If aP = mod…

A: Given the data

Q: Suppose that m is a positive integer. Use mathematical induction to prove that if a and b are…

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Q: Let a and b be positive integers with (a, b) = 1. Show that if g is a primitive root (mod ab), then…

A: We have given (a,b)=1 where a, b both are positive integer. We have to show if g is primitive root…

Q: Consider the congruence mod 6 for integers. Z6={[0], [1], [2], [3], [4], [5]}. Prove the product of…

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Q: Suppose a is a primitive root modulo n. Prove that the set {a, a², ... ,a*(n)} is the set of all…

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Q: 6. If c is a positive integer and a and b are integers with a = b(mod c), show that a2 = b²(mod c).…

A: Let c is positive integers and  a and b are integers with a ≡ b ( mod c)

Q: Let m1, m2,…, mn be pairwise relatively prime integers greater than or equal to 2. Show that if a ≡…

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Q: The equivalence relation x = y(mod4) is defined on the set Z of integers. Determine the quotient…

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Q: Consider the congruence mod 6 for integers. Z6={[0], [1], [2], [3], [4], [5]}. Prove the sum of an…

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Q: Let m be an integer with m > 1. Note that a = b (mod m) if and only if m divides a – b. Define the…

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Q: In this exercise set, all variables are integers. 1. List the distinct congruence classes modulo 5,…

A: We can solve this using linear congruence method

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

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Q: Show that if a, b, and m are integers such that m > 2 and a = b (mod m), then ged(a, m) = gcd(b, m).

A: This question is related to Number Theory

Q: Let 1Sasb be two positive integers. How can I prove or disprove that there exists a 1sks b-1 such…

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Q: Let p be a prime such that p mod 4 = 3. How many elements oforder 4 are in GF(pn)*?

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Q: In the ring of integers modulo n, (Z„ +, ·) prove that m e Z, is a zero divisor e (m, n) > 1.

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Q: Label each of the following statements as either true or false. Let A be a set of integers closed…

A: Given: Let A be a set of integers closed under subtraction. If x and y are elements of A, then x-ny…

Q: Use the definition of congruence to prove that if j, k, 1, m, nE Z are such that k =1 (mod n) and…

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Q: Prove that if p is a prime number then Z mod p is an integral domain

A: Integral domain:- A commutative ring R with a unit element 1 with no zero divisors is said to be…

Q: Given that a, b, c,m are integers with m > 0. If a = b (mod m) then prove that ас3 bе (mod m)

A: we have to use the only definition of congruence and the definition of divisibility

Q: Prove, for all integers x, if x mod 3 is not 0, then 3|(x2+2).

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

Q: 2.1.3. If a, b are integers such that a = b (mod p) for every positive prime p, prove that a = b.

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Q: Prove that for any integer a, at = 0 (mod 5) or aª = 1 (mod 5)

A: To prove for any integer a,  a4≡0mod5 or a4≡1mod5.

Q: 4.18. Let m, ń EN such that m|n. Prove that if a and b are integers such that a = b (mod n), then a…

A: The detailed solution is as follows below: