(7) Let m≥ 1 and let R= {0, 1, 2,...,m - 1}. (b) Prove that for every integer a there is a unique integer r ER such that a = r (mod m).

Q: 8. Let m 2q + 1 be an odd positive integer. Prove that every integer = is congruent modulo m to a…

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Q: let a and b be two relatively prime integers. if there exists an m such that (a/b)^m is an integers,…

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Q: Define L: Z → Z and M: Z → Z by the rules L(a) = a² and M(a) = a mod 5 for each integer a. (a) Find…

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Q: Prove that the analogue of lemma 1.23 is not true for numbers of the form 4n+3 where n is an…

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Q: Let a be an integer greater than 0 and b be an integer greater than or equal to 0. Let a be greater…

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Q: 1) Compute the order of a modulo 7 for a = 1,2,....6. Find all primitive roots modulo 7.

A: Order of the element

Q: do I prove this: Let a, b ∈ Z. Give an example of an integer x so that x∣a and x∣b and If x∣a and…

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Q: Let x E Z. Prove that either x2 = 0 (mod 4) or x2 = 1 (mod 4).

A: By the help of even and odd integer's we prove the result. The answer is given as follows :

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

A: I hove used the theorem on congruent

Q: (a) Prove that 2 and 3. are both primitive roots modulo 5.

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Q: Prove that o – 1 is divisible by 4.

A: Even numbers are numbers divisible by 2. Odd numbers are numbers that are not divisible by 2. The…

Q: Prove that an odd integer n is a square modulo every power of 2 if and only if n ≡8 1.

A: Given n is an odd integer. We have to prove that an odd integer n is a square modulo every power of…

Q: (6) (a) = Let m be a positive integer and let a be an integer such that gcd(a, m) = 1. Prove that…

A: (a)Given that: m and a are positive integers and,.This implies that there exist integers u and v…

Q: (b) There exists an integer b such that for all integers n > b, n³ – 8 is prime.

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Q: Let m 2q be an even positive integer. Prove that every integer is congruent modulo m to a unique…

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Q: Let n and k be integers such that (n, k) = 1. Prove that (n, n − k) = 1.

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Q: Prove that for every positive integer n, Q/Z has an element of order n.

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Q: Prove that n < 2n for all n ∈N

A: We prove n&lt;2n by principle of mathematical induction Principle of mathematical inductionStep 1:…

Q: Prove that for any integer x, if x mod 3 does not divide 0, then (x2 −3x + 1) mod 3 = 2.

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Q: Let 1 ≤k≤p-1 where p is a prime number. Then prove that (₁7¹) = (-1) (modp) k

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Q: Let a and b be non relatively prime positive integers. If there exist integers s and t with as+bt=2…

A: From the result, if a and b two integer and GCD(a,b)=d then there exists two integers x,y such that…

Q: 4) Determine if the following statement is true and either prove it or provide a counter example:…

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Q: Prove that if a1, a2, . . a, are n > 2 integers such that a; = 1 (mod 3) for every integer i (1 < i<…

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Q: a) Let a be an integer with a > 0. The integer a is even if and only if a = 0 mod 2. Note to self:…

A: (a) Let a be an integer with a &gt; 0.       We need to prove that a is even if and only if a ≡ 0…