Suppose that a, b, m, and n are integers (n, m positive) and suppose we know a = b (mod n) and n | m Can we safely conclude that a = b (mod m)? (a) True by transitivity (b) False because I can find small integers for which it fails. True, because congruence mod n has more equivalence classes so the conclusion is just a weaker (c) version of the hypothesis. True, because congruence mod n has fewer equivalence classes so the conclusion is just a weaker O (d) version of the hypothesis.

Q: Show that if a and b are integers and m and n are positive integers such that n|m, then If a = b(mod…

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Q: Prove that if ca = cb (mod n), then a = b (mod n/d) , where d = gcd(c,n). %3D

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Q: 5. Prove that m(n) + nº(m) = 1 (mod mn) if (m, n) = 1. %3D

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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: Prove that for all integers x, if x is not divisible by 7, then x^3 ≡ 1 mod 7 or x^3 ≡ −1 mod 7.

A: Since you have asked multiple questions, we will solve the Question Q2 for you. If youwant any…

Q: to show that the numbers in question are incongruent modulo n.] 11. Verify that 0, 1, 2, 22, 23,…

A: We shall solve this question in next step.

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Q: 24. Prove that for all integers m and n, if m mod 5 = 2 and n mod 5 = 1 then mn mod 5 = 2.

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Q: Let n be a fixed positive integer greater than 1. If a mod n = a' andb mod n = b', prove that (a +…

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Q: Number theory

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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: 8, Suppose a,b € Z. Prove that a = (mod 5). b (mod 10) if and only if a = b (mod 2) and a = b

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Q: Let P be a prime, if ac = bc (mod p) and p does not divide c, then a = b (mod p) Select one: O True…

A: Given  gcd(c, p) = 1 and ac ≡ bc(mod p). (We may assume p > 1 so that c  ≠  0.) Then ac − bc =…

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

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

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Q: 1. Suppose m, nEz* are relatively prime. Prove that for all a, bEZ, a =b (mod mn) if a =b (mod m)…

A: Note that: a is congruent to b modulo m if and only if m divides (a-b).

Q: Prove that if x2 is congruent to y2 (mod n) and x is not congruent +or - y (mod n)  then gcd(x + y,…

A: Given: x2≡y2modn x≢±ymodn

Q: Suppose x satisfies the three simultaneous congruences X = 4 (mod 5) x = 7 (mod 8) X 10 (mod 11)…

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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: Prove that for all integers x, if x is not divisible by 7, then x^3 ≡ 1 mod 7 or x^3 ≡−1 mod 7

A: We will prove the given statement.

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: Suppose that a1 ≡ b1 mod (m) and a2 ≡ b2 mod (m) -Show that a k 1 ≡ b k 1 mod (m) (hint: use…

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Q: Suppose that (n, e) is an RSA encryption key, with n = pq, where p and q are large primes and gcd(e,…

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Q: (b) Prove that multiplication mod n is commutative.

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Q: 30. Prove that any positive integer is congruent to its units digit modulo 10. 31. If a = b (mod n),…

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Q: b) Prove that if n2 (mod 3) = 1 then n is not divisible by 3.

A: The required proof is as follows -

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: 1. Let p and q be distinct prime numbers and let a be an integer not divisible by p and not…

A: As per BARTLEBY GUIDELINES , I answered one question (i.e. Q-1) and repost other questions…

Q: Let p >2 be a prime. Prove that [(p-2) = (-1)/2 (mod p).

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Q: Use induction to prove for every positive integer, 4 x 10n = 4 or 7 (mod 33)

A: Suppose P(n) be any mathematical statement in terms of n. Then Principle of mathematical induction…

Q: Prove that the relation of congruence modulo n is transitive. That is, prove that for all integers…

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Q: (b) Prove that if p is a prime greater than 2 then 2(p- 3)! = -1 mod p %3D

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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 if n is an odd positive integer then 1 + 2 + 3 + · · · + (n − 1) ≡ 0 mod n . What happens…

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Q: Show that if a = b mod m and b = c mod m, then we have a = c mod m. Use the definition of congruence…

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Q: 5. By the principle of nathematical inductionprove that 3antl +(-D"2=O(mod 5) forall positive…

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Q: Prove or disprove: for any integers a, b, c, d, if a= c (mod n) and b= d (mod n), then ab = cd (mod…

A: Given: For any integers a, b, c, d if a≡cmod n and b≡dmod n To show: ab≡cd (mod n) To say that a is…

Q: Recall that the set of integers mod n, Z/nZ, consists of the n equivalence classes under =. For…

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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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A: The given statement is False.  Consider, x = 5 and d = 4. Then we see, x2 - 3x = 52 - 3x5…

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A: consider an integer n=26 such that n mod 6=26 mod 6               =2 mod 6 also, n mod 9=26 mod 9…

Q: Suppose that Bob has used two distinct primes p and q to create the public modulus n = pq and public…

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