Let p be prime. Suppose ab≡0 mod p and a ≢ 0 mod p. Apply Euclid’s Lemma to prove b ≡0 mod p

Q: 6. Let m, n EN with m, n > 2, and let a, b E Z. Prove that a = b (mod mn) implies a = b (mod m) and…

A: (a) a = b (mod mn)       => there exists some integer k such that                    a = b +…

Q: Let n E Z+, n > 2. Show that n is prime if and only if (n – 2)! = 1 (mod n).

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Q: Prove that : If p is a prime number and p does not divide a then a- 1(modp).

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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: Let p ≥ 2 be a prime. Suppose p does not divide a. Then ap−1 ≡ 1 (mod p).

A: The given problem is to show the given following modular congreunce that let p is a prime and p…

Q: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p= 1 (mod 4)

A: Compute general prime that satisfies given condition as follows.

Q: Use induction to prove that for every n E N, we have 5" = 1+ 4n (mod 8)

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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: consider the statement: for all n∈N , x∈Z , x^2 ≡ 1 mod n (        ) x ≡ 1 mod n or x≡ -1 mod n a)…

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Q: Let p be a odd prime number and b, c € Z. Assume that there exists l e Z, such that l² = b² – 4c mod…

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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: Let a, b E Z and n E N. For each statement decide whether it is necessarily true, or whether it can…

A: The given problem is to prove the following identities, we have to use basic identities of modular…

Q: Prove or disprove If ac = bc(mod m), then a = b(mod m). If n| m and a = b(mod m), then a = b(mod n).…

A: Hello. Since you have posted multiple questions and not specified which question needs to be solved,…

Q: Prove that if a and b are integers, then (a^5 + b^5) is congruent to a^5 + b^5 modulo 5.

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Q: Consider p> 2 a prime number, justify that: (P,') = (-1)" (mod p) to any 0 sn sp- 1,n EN. 1.1 1.2…

A: In the given problem we use the definition of combination. Then the problem is solved by…

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

A: The solution is given as follows

Q: 3. Use the above theorem to prove that if p E N is a prime number such that p = 1 mod 4, then p is…

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Q: Assuming Wilson's Theorem, prove that if p prime and (a, p) = 1, then aP-1 = 1 mod p.

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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: 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: Prove a2 = c² (mod n), assuming that a, c and n are integers with n > 1 and that a = c (mod n)

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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: Suppose p is an odd prime number, prove that p+1 (р — 3)! %3D (mod p) -

A: To prove: To prove that p-3!≡-p+12(mod p) for an odd prime number p.

Q: If p is an odd prime, prove that: 1² · 32 . 52 ... (p – 2) = (– 1)P*1)/2 (mod p) and 22 . 42 . 62…

A: Given that p is odd prime. (i) To prove 12.32.52....(p-2)2≡-1p+12modp Since,…

Q: Let a, b e Z and n E Z*. Prove that if a = b(mod n), then gcd(a, n) = gcd(b, n).

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Q: Give an example to show that a? = b? (mod n) need not imply that a = b mod n). fa=b(mod n) prove…

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Q: Q1 Prove or disprove i. If ac = bc(mod m), then a = b(mod m).

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Q: Let p be prime. (a) If a 1 (mod p), show that 1+ a + ...+ aP-1 = 1 (mod p). (b) If a -1 (mod p),…

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Q: Prove the following statement or disprove it by giving a counterexample: "For all positive integers…

A: The given statement, "For all positive integer n and for all integers a and b, if a2≡b2modn then…

Q: If a has order n-1 mod n, then n is prime. O True O False

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Q: Let p > 3 be a prime. Show that if p2+ p + 1 is also prime, then p =5 mod 6.

A: We prove the result by the method of contradiction.

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: If a is an integer such that: 2 ła, 5ła Show that (a)⁴ⁿ = 1(mod 40) for any positive integer n

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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: Let a,b,and c be natural numbers and n be a natural number > 0. Prove that if a = b mod n, then a^c…

A: Use mathematical induction to prove the given statement.

Q: Consider numbers of the form 9" mod 13, where n E Z+. We claim that all of these numbers will only…

A: First principle of mathematical induction : Suppose Pn is a statement where n∈ℤ+ . The statement is…

Q: Let n be an integer. Prove that, if n mod6 = 2, then n2mod12=4

A: We apply concept of modulus here to solve this.

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: e) Let p and q >7 be primes such that 2 +2P +1 =0 mod q. Prove that q = 1+6kp for ome natural number…

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Q: Prove or disprove. There is a prime of the form p = 6k – 1 such that the congruence 13x? + 7x +1 = 0…

A: We have the given Quadratic function is f(x)=13x2+7x+1 The Discriminant is given by…

Q: Let n E Z. Prove that 3n # 1 (mod 9).

A: Proof by contradiction: Direct approaches can be difficult (or impossible) to use to prove a…

Q: Prove that if a = b mod n and c= d mod n, then ac bd mod n.

A: Given if a≡bmodn and c≡dmodn We have to show that ac≡bdmodn. Now, a≡bmodn…

Q: Let a E Z. Prove that if a = 2 (mod 6), then a? = 4 (mod 12).

A: We will prove this by basic result of congruence

Q: Show that for x and y integers, and m a positive integer x = y mod m implies x = y mod m.

A: This is a problem of Number Theory, Modular Arithmatic.

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: 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: For n=195, by Fermat's theorem d-2 a) a (modn) b) 1(modn) c) 0(mod n) d) a(mod n)

A: Given:                 n=195 To find:                an-2=? by using Fermat's theorem Fermat's…

Q: For n=195, by Fermat's theorem a-2 a) a' (modn) b) 1(modn) c) 0(modn) d) a(mod n)

A: Fermat's theorem

Q: (a) Consider the following statement: For all n € N, x(x − 1) = 0 mod n Which of "if", "only if",…

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Q: Prove the above property. That is, prove that if a mod n = k and b mod n = j, then (a + b) mod n =…

A: If a mod n =k and b mod n=j show that a+b mod n =j+k mod n. Definition: If x mod y=z, then y | x-z.…