(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: Prove or disprove For every integer m, m(m+1) is even

A: To prove or disprove the given statement- For every integer m , m(m+1) is even .

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 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: Prove that (1 + x)n≥ (1 + nx), for all natural number n, where x &gt; – 1.

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Q: prove that for all integers m, n & d if d/m & d/n then d/(2m+3n)

A: We know that if a and b are integers such that a/b then b is a multiple of a. i.e. b=ka where k∈ Z{…

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

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Q: Prove that 25|(6" – 5n – 1) for all positive integers n. -

A: We can prove given statement by principle of mathematical induction: Alogrithm: Obtain P(n) and…

Q: Let k be a positive integer. Show that 1" + 2 + 3™ +....+(n – 1)m + n™is O (n™+1). -

A: According to the given information, it is required to show that:

Q: Let a and b be natural numbers, where a,b >= 2, such that gcd(a,b)=1 and ab=(x+y)n for some natural…

A: Given that a and b are natural numbers and  a , b ≥ 2 .  Also, given that gcda,b = 1 and ab = x+yn…

Q: Show that if a|m and a|(ms+nt) for some integers a≠0, m, s, n, and t, then a|nt.

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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: b) Prove that () - 2) = n², where n is a positive integer.

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Q: Prove that the fourth power of any integer has the form 4k or 4k + 1 for some integer k.

A: We have to prove the following -  The fourth power of any integer has the form 4k or 4k+1 for some…

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: Prove: If m and n are non-zero integers and m > n, then there exists an integer q such that (1/n -…

A: Let m and n are non-zero integers such that m&gt;n To prove there exists an integer q such that…

Q: Let a and b be relatively prime positive integers. Show that a(b) + b(a) = 1(mod ab).

A: Solution:

Q: Let k and n be two positive integers, and assume that n is odd. Prove that there exists an integer a…

A: Given: Let k and n be two positive integers in that, we assume if n is odd.

Q: Prove that for every ponitive integer a if aº – 6° is a prime for some positive integer 6 5 = 2 or 5…

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

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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…

Q: Prove: If a is a positive even integer and b is an odd integer, then (a,b)=(a/2, b).

A: Let d be gcd of a and b. Now, b is odd so d cannot be even or is odd.  Now, since d is odd and…

Q: Negate the statement: For all positive integers, n, there exists a positive integer, k, so that…

A: The negation of a statement is:- Statement:- p implies q. Negation:- p does not imply q.

Q: Prove that for all integers n, [n/7] = (n – 3)/7 if and only if n mod 7 = 3.

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Q: Prove that 1 n i(i+ 1) n+1 i=1 whenever n is a positive integer.

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Q: Let A = {1, 2, ..., 12}. Let aRb mean a = b (mod 5). (a) Give [2]R. [3]R, and [5]R. (b) Give [2]…

A: A set of ordered pairs is referred to as a relation in the area of mathematics and computer science.…