Suppose n is a large odd number. You calculate 2(n-1)/2 = k (mod),  where k is some integer k is not congruent to +/- 1 (mod n). a) Suppose k2 is not congruent to 1(mod ). Explain why this implies that n is not prime. b) Suppose k2 is congruent to 1(mod ). Explain how you can use this information to factor n.

Q: TION 9 Prove by Induction that for every natural number n, 3"–1 is a multiple of 2? Attach File…

A: We prove by induction

Q: Assume that k mod 15 4. Compute 11k mod 15. Show all your work clearly.

A: Topic:- basic maths

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A: Ans: -87

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Q: a) Prove that 10" =+1 (mod 11) for any n EN. b) Suppose the integer r has digits rI-1I, I0. Prove…

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Q: Prove: in L number psiz 12, for every prime poiz, [p] = [1], [5) , [7) , or [11]

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Q: What is the least residue of 7162 modulo 17?

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Q: Suppose n > 1 is a composite integer ab where a and b are unequal integers both greater than 1.…

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Q: Number theory

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Q: 8. Use Euler's Theorem to find a number a between 0 and 9 such that a is congruent to 7 1000 modulo…

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Q: find the integer x such that 0 < x < 232 and 144x = 4(mod 233). Show your work.

A:

Q: Is it true that 90 ≡ 64 (mod 13)?

A: As given equation 90≅64(mod 13) is False.

Q: Characterize the primes p mod 20 such ial

A: We have to characterize p mod 20 such that -5p=1.

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: 122 is congruent to 5 modulo17 ?

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Q: Is 7|(15+20): True or False? 28 is congruent to 42 modulo 6: True or False?

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Q: To Determine the value that 46 is congruent to IN modulo 7

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Q: 11. Define conguren ce class es for modulo 4.

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Q: Using Fermat's theorem, find 3201 mod 11.

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Q: Prove that if a + x ≡ a +y (mod n), then x ≡y (mod n)

A: I have proved it by using defination of congruent.

Q: Prove Exercise 4.6.12 in the course notes. Then, assuming that gcd(a, n)|b, determine how many…

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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: Determine whether 17 is congruent to 5 modulo 6?

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Q: Use Wilson's Theorem to prove that for primes p congruent to 1 modulo 4 the quadratic congruence 1'…

A: Given The quadratic congruence x2≡-1(mod p) To prove  x2≡-1(mod p) has a solution if p≡1(mod 4)…

Q: Suppose n is a large odd number. You calculate 2(n-1)/2 = k (mod),  where k is some integer k is not…

A: Given that 2n-12≡kmodn Take the square on both sides, 2n-12≡kmodn2n-12·2≡k2modn2n-1≡k2modn (a) The…

Q: Use a direct proof: p is a prime # and x ∈ Z. If x2≡x (mod p) then x ≡ 0 (mod p) or x ≡ 1 (mod p).…

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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: Find the least residue of 99! (Mod 101)

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A: To show it by Euler's theorem :-

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

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.