Recall from Student Exercise 1.4.8 that cos 8 ² ( ²53 ) = 1/²/ ( √ √² + √²) (√₂ 70 Written Assignment 3, we showed that cos = Use mathematical induction to show that in fact, for all n ≥ 2, ㅠ 1 0² (2) - ( √2 + √2 + + √2 COS = 2n n-1 2 In √2+√√2+√2

Q: Consider the linear congruential number generator given by n + (ахр-1 + с) mod m

A: In this question, we have given a linear congruential  generator xn←axn-1+c  mod(m)Given thata=4,…

Q: In GF(2), find the inverse of x^3+x+1 modulo x^5+ +x^2+1. Show your complete sölution Oax^3+x…

A: For some finite field  F and fx ,gx∈Fx s.t gx≠0 On applying successive division algorithm:…

Q: 3. Prove that ox + 6x -4X12 is irreducible over Q.

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Q: (d) (x² + 2x + 1)q(x) = 1 (mod x³ + x² + 1) over Z3

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Q: Need help modern algebra defining D_2n and showing order D_2n=2n

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Q: 13. Floor and ceiling notation were defined in Section 1.1. (a) Show that []+[=n for all positive…

A: Floor and ceiling.

Q: Let p be a prime that does not divide n. Prove that Φpn(x) =Φn(x p)/Φn(x).

A: Given: p be a prime number that does not divide n. To Prove: Φpnx=Φnxp/Φnx.

Q: Theorem 4.12. Gauss: Let n #0 (mod p). Consider the remainders mod p of the following (p-1)/2 many…

A: Let n≢0mod p. Consider the remainders mod p of the following p-12 many integers, n, 2n,......,p-12n.…

Q: If P(A)=0.2, P(B)=0.3 an d :P(ANB)=0.1, then P(A'UB) is

A: As per bartlby guidelines we can answer only first question and rest can be reposted  Given:…

Q: (b) Let Ho =1, H1 =3, H2 =9, and define Hk = Hk-1 + 3Hk-2 + 3Hk-3 for k23. Show by induction that Hk…

A: We will prove the result by induction  H0=1≤30 H1=3≤31 H2=9≤32 The result is true for k=0,1,2 Assume…

Q: 1.13 Let the numbers 9₁,92, 93,... be defined by 91=2, gn = 3qn-1 - 1 (n ≥ 2). Show by induction…

A: Mathematical induction is a method of solving mathematical problems. This method has 3 steps:(1)…

Q: (11.1) Show that for any two positive integers m and n. m {m² + m + ¹} = {x{"+ *} m k k=0

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Q: Fn denote the nth Fibonacci number.

A: Given,              A=1110 To prove that,              An=Fn+1FnFnFn-1

Q: Let m be the number of these remainders which exceed p/2. Then ( #) = (-1)m.

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Q: Q3 Not submitted Section 4.1 # 14 Show that if a, b, and e are integers with e > 0 such that a = b…

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Q: 5. Let f(x) = 2x² + x + 4. It is known that f(1) = 0(mod 7). (i). Find a solution to f(x) = 0(mod…

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Q: By subtracting (17) from (16), we get C (P− Q)[(A+B+C+1) −D] =0, Since (A+B+C+1) − D ‡0, then P = Q.…

A: We have been given: P=A+B+CQ+DP-be-d    16Q=A+B+CP+DQ-be-d    17 We have to explain: P-QA+B+C+1-D=0

Q: Let n and k be integers such that (n, k) = 1. Prove that (n, n − k) = 1.

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Q: Show that every integer of the form (6m + 1)(12m + 1)(18m + 1), where m E Z+ such 6m+1, 12m+1, and…

A: Carmichael number is an odd composite number which satisfies the  congruence relation  bn-1=1(mod n)…

Q: 1. a) Show that 2 is a primitive root modulo 19. b) Determine whether 5 is a primitive root modulo…

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

Q: 4. Let the function g be given by Oniseven 9(n) n =1 (mod 4). n =3 (mod 4) 1 %3D 1

A: This is a question from Number Theory.

Q: brvib - If m is composite, what is the value (m – 1)! mod m in the standard residue system? Conclude…

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Q: (a) Prove the following: i. 11 + (6 x 4") - 22n = 6 mod 7, 1 is divisible by 131, ii. 8260 iii.…

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Q: Find all solutions of the congruence 3 f(x) = 5x³ - 93= 0 (mod 231).

A: The given congruence equation is .We need to find all the solutions of it.

Q: (a) Find an r E Z that makes the following congruence true: 5x = 1 (mod 24). (b) Prove that there is…

A: Solution a: Given  5x≡1 mod 24 For x=1 5≡5 (mod 24) for x=2 10≡10 (mod 24) for x=3 15≡15 (mod 24)…

Q: 6. Let Fn 22" + 1, n > 1, be a prime Fermat number. Show that 3 is a primitive root modulo Fn.

A: Given: Fn=22n+1,n>1

Q: Recall the Fibonacci numbers Fn are defined by Fo = 0 , F1 = 1 and Fn Fn-1 + Fn-2 for all n > 2 .…

A: Consider the provided question, Fibonacci numbers Fn are defined by F0=0, F1=1 and Fn= Fn-1+Fn-2 for…

Q: Let a = (1 7 3)(5 94 2) and B= (2 3)(7 4)(5 1 8) be elements of S10. Then, a B is O (1 4)(2758 9 3)…

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Q: Show that — ф+1 n=0 ||

A: Given φ is the golden ration satisfying 1φ=φ-1. To prove ∑n=0∞1φn=φ+1 It is known that φ=1+52.

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: b If U = x + express x° + 1 m the form U + AU4 + BU² + C. State the values of A, B and C.

A:

Q: Theorem 5.3. Let a be an odd integer. If a3 we have (29) a 2 = 1 (mod 2º), so there are no primitive…

A: Proof:                 We shall prove by induction on α. First we consider α=3. Let α be an odd…

Q: Let Z3[i] = {a + bi : a, b ∈ Z3, i2 = −1} where arithmetic is done mod 3. 1.  Is Z3[i] an integral…

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Q: Determine whether e(x)=15x+4(mod 26) and e(x)= 4x+5(mod 26) are invertibles or not. If yes, find the…

A: We are asked to determine if the given equation are invertible and if they are we have to find the…

Q: 3. (i). Suppose f (x) is a polynomial with integer coefficients such that f(2) = 0(mod 5) f(4) =…

A: Given that  (a) f(2)≡0(mod 5)f(4)≡0(mod 9) We have to find a polynomial such that f(x)≡0(mod 45)…

Q: 2. Show that 1184 – 534 is divisible by 7. -

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Q: 1. a) Are the following statements true or false? Statement 1: Every integer x is of the form 2k or…

A: Solution:

Q: 10. a) Recursively define ao = 1, a; = 3, a2= 5 and an = 3an2 + 20ars for n2 3. Çalculate.gafor n =…

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Q: 18.) Compute fg (Note: each should only be one In 2₁, we write -1 as f(x)=x²-3x +2, term in the…

A: fg(x) = f(x)g(x)

Q: Prove that "Co "C₁ "C₂ + 22 3² 1 ... + (-1)". 1 n+1 "C₁ (n + 1)² 1 + 1 1 2 + 3 + + 1 n+1

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Q: 6. If |G| = pq for distinct primes p and q, show that G is not simple. (Note the slight difference…

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Q: duce the expression for u to 2(5 ,Y = (1+5P) +4 (mod 2p - 1) or simply, to 2u, = 1+5P-1(mod 2p - 1).…

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Q: Which of the following would be a valid post-condition of the following Hoare triple? x = = (i+1)!.…

A: Answer  Option C is correct