6) If p is an odd prime, then 1P + 2P + 3P + .. + (p – 1)P = ??(mod p) a) 0 b) 1 c) - 1 d) None of the above

Q: *4. Prove that 1³ +2³+ ... + n² = n²(n+1)² for all ne N.

A: Introduction: A strong and elegant method for proving specific kinds of mathematical claims is…

Q: Show that if n is an integer and 3n3 + n is odd then n is odd, by using a) Proof by Contrapositive…

A: To prove i - if n is integer and 3n3+n is odd then n is odd a) By contrapositive :   (P→9)≡(~q)…

Q: (a) Show that if n 7 mod 8, then n + 6n 3 mod 8.

A: If, n = 7 (mod 8) So, n² = 7² (mod 8) So, n² = 49 (mod 8) So, n² = 1 (mod 8)

Q: ind witnesses to show that n^3 − 5n^2 − 4n is O(n^3).

A: In this question, order of polynomial can be found with the help of Big-O notation. Big-O Notation…

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Q: (3) (2p = 2 (mod p), where (2) n! for 0 <k <n. k!(n - k)!

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Q: If p is a prime with p=3 mod 4 and a is quadratic residue mod p, then -a is a quadratic residue mod…

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Q: If p is a prime with p = 5 mod 6, then (²) O -1 O None of these

A: If p≥3 is an odd prime, then    (-3/p) = 1 if p ≡ 1 (mod 6)    (-3/p) = -1 if p ≡ 5 (mod 6)

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Q: What is the smallest positive integer k such that xk = e for all x inU(7 . 17)? Generalize to U(pq)…

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Q: 15 Composite Then and O modn Show that if n there are with X#0 od n and y7 but where yy= Omod n.

A: Given n is a composite number

Q: prove that the following integer is composite (not prime). Do not use mod. 2^35(2^33-1) + 1

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Q: (5) Show that if n ɛ Z+ is odd which is divisible by k distinct primes, then the congruence x2 = 1…

A: Chinese Remainder Theorem: If a1,a2,...,ak are integers, and m1,m2,...,mk are pairwise relatively…

Q: (a) Show that, for any odd prime p > 5, 1 if p = 1, –5 (mod 12) (주) -1 if p= -1,5 (mod 12)

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Q: 2. Prove by contraposition: For all integers n, if (3n +4) is even, then n is even.

A: Contrapositive of 'if p then q' is if not q the not p'

Q: Show that if n is an integer and 3n3 +n is odd then n is odd, by using a) Proof by Contrapositive b)…

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Q: 24. Show that 1P + 2P + 3P + .+ (p - 1)P =0 (mod p) when p is an odd prime. Show that if nis prime…

A: Using Wilson's Theorem, Fermat's Little Theorem and The Pollard Factorization Method, we have to…

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

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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: 6. Show that ifn = pq, a product of distinct primes, then a (n)+1 (mod n) for all a. = a

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Q: Establish that if a is an odd integer, then 2" а = 1 (mod 2" +2) for any n 2 1.

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Q: If p>=q>=5 and p and are both primes, prove that 24| p^2-q^2.

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Q: 24. Show that if p is a prime and k is an integer such that Isksp-1.then p divides ().

A: See the handwritten solution

Q: Suppose that m, n are two integers such that m 2n2 = 1. Does it follow that m and n are relatively…

A: Given that m, n are two integers such that m3-2n2=1. To show m, n are relatively primes we have to…

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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: (b) Prove that if a = 7 mod 10 and b = 6 mod 15, then a + b = 3 mod 5.

A: Our objective is to prove . Since you have posted multiple questions I am solving one.

Q: Suppose that m, n are two integers such that m3 – 2n2 = 1. Does it follow that m and n are…

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Q: . show that for every positive integer n, if n^2+2 is prime then n is a multuple of 3. Is the…

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Q: 2) Find the smallest positive integer z, such that z = 13 mod 4 z = 20 mod 9 z = 11 mod 25 .

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Q: 2. Prove that if n is even, then 3n + 1 is odd

A: The given expression is 3n + 1. Prove that 3n + 1 is odd, if n is even.

Q: (4) Let p be a prime number. Then r = 1 mod p has exactly two solutions modulo p.

A: 4) To prove, x2≡1mod p has exactly 2 roots such that p is a prime number.

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: 15. Prove that for each natural number n, s+ (4n-3)(4n+1) 4n+1 ...

A: Given The given expression is 11·5+15·9+·  ·  ·+14n-34n+1=n4n+1

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: in Z12. L12, for every prime number p>iz, [p] = [1], [5], [7) , or [11]

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Q: Prove that if a > 0, k > 1, and ak -1 is prime, then a = 2 and k is prime. %3D

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Q: Show by using construction that n^2+5n+6 is not prime for any n is a natural number.

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Q: (i) Prove that there are infinitely many primes of the form p = 6n – 1. -

A: Given:    Infinitely many primes of the form p=6n-1     Proof:         Suppose p1, . . . , pn are…

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: If n and a are prime to 7 then prove that nº – a® divisible by 7?

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

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Q: Suppose that m,n are two integers such that m3−2n2=1. Does it follow that m and n are relatively…

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Q: 3) Support the p is a that Show least one Prove then iss prime 27. ท (1^2) = (2+1) = 1 for at number…

A: Disclaimer:  “Since you have asked multiple question, we will solve the first question for you. If…

Q: (a) gcd(q, n) = 1 if and only if kq # 0 mod (n) for all 1 < k< n. %3D

A: