Discrete Math (a) Solve the congruence relation 3x + 5 ≡ 2(3 − 5x) (mod 35) for x. (b) If a ≡ b (mod n) and c ≡ d (mod n), show that ac ≡ bd (mod n). (c) Z∗ 39, the set of all the reduced residue classes modulo 39. (d) φ(39), where φ is the Euler φ-function.

Q: 3: Suppose gEF is a generator and j,d€Z. (a) Please prove that gid=1 mod p if and only if jd=0 mod p…

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Q: Let p(x) be a polynomial with integer coefficients, and let n e N. Given two integers a and b such…

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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: Proof Assume that 2 | (x² − 1). So x² − 1 = 2y for some integer y. Thus, x² = 2y + 1 is an odd…

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Q: 10. Using Cauchy's Residue Theorem, evaluate -2π (cos 30)² 1-2p cos 20+p² d0,0 < p < 1.

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Q: • 2π 0 1 1 + cos² t dt

A: ∫02π11+cos2tdt

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 TE A (V) and if 9 (1) € Fox) is such that a (T) = 0, is a characteristic root it true that every…

A: The statement is false.

Q: (c) Find a value t modulo 7 so that (3+ 7t) does not solve the congruence x® – 1 = 0 (mod 49).

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Q: Evaluate by the residue theorem: cos 20 de 1-2p cos 6+p2: 2T p<1. 1-p².

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Q: Find the order of the residue class 5 in Z/29Z.

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Q: Suppose that the prime factorizations of two positive integers x and y are expressed using a common…

A: The prime factorization is breaking down a number into the product of only prime numbers. These…

Q: Let 1 ≤ m ≤ n be integers. Give a combinatorial proof of the identity n+m Σ (:) ( ".) ( ^_) - (.")…

A: We have to proof ∑k=n-mnnknn-kkn-m=nn-mn+mm Let, there are two group X and Y and each group having n…

Q: Evaluate dr (1+*²)² by the residue theorem: So [#].

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Q: J in =d [ Å ] ∞

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Q: 1. (i) Find all four square roots of 61 in Z247. (ii) Show that r³7 = x for all r E Z217- (iii) In…

A: (i)      Let 'a' be square root of 61 in Z247 Then,       a2   = 61 (mod 247)     i.e.,  a = 247 K…

Q: Let g(x) 3 +x+7. = X i) Find a solution to g(x) in Z/21Z. ii) Find a solution to g(x) in Z/3969Z…

A: We will use the basic knowledge of polynomials and the elementary theory of the congruence modulo…

Q: 4. Find the degrees of the following extensions. (a) Q(√3) over Q. (b) Q(√3, i) over Q(i). (c)…

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Q: Theorem 6 If (b+f) > (c+r) and (d+ g) > (e+ s), then the necessary and sufficient condition for…

A: Given : P=aQ+b+fP+c+rQd+gP+e+sQ,   Q=aP+b+fQ+c+rPd+gQ+e+sP

Q: . Let f (x) = 11x +9x +21. Prove that f (x) is irreducible over Q.

A: We solve this problem using  (mod p ) irreducibility test over f(x) where p = 2 , a prime number.

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: ami 3 Find explicit formula for ao = 1, a₁ = 2, an = an-₁ + 2 an-2, cima characteristic polynomial…

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Q: 2.3. Let g be a primitive root for Fp. (a) Suppose that x = a and x = b are both integer solutions…

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Q: Evaluate by the residue theorem: So √(1+2+7²) [#]. dz

A: Given:The integral To find:The integral using residue method.

Q: (a) Determine whether 2 is a primitive element in Z37. (b) Find an element of Z37 that has order 12.

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Q: Evaluate ·∞ In z de 4+x² 500 by the residue theorem: [*²].

A: The given integral is: I=∫0∞ln xx2+4dx Let fz=ln zz2+4. So, the poles are where: z2+4=0z2=-4z=2i,-2i…

Q: (b) Show that if gcd(m, n) = 1, then σ₁ (mn) = σt(m)ot(n). In other words, show that function. Is…

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Q: Yn+1 = ayn-p+YYn-h (yn-h) 9-¹+8 +6

A: Given: The information showing the proof of a specific theorem.To prove: yn+1=αyn-p+γyn-hyn-hq-1+δ…

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: Theorem 6 If (a1 + a3 + a5) > (a2 + a4) and (B1 + B3 + B5) > (B2 + B4) , then the necessary and…

A: Given, if (α1+α3+α5)&gt;(α2+α4) and (β1+β3+β5)&gt;(β2+β4), then the necessary and sufficient…

Q: Determine a and b such that p(x) = x² + ax + b satisfies p(1) = 0 and p'(1) = 9. (Give an exact…

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Q: (c) Show that V3 e Q[ V6]

A: We have to proceed using contradiction method.

Q: [(@1 + a3 + a5) - (a2 + a4)] [(B1 + B3 + B3) + A (B2 + B4)]' P+Q = (4.17) %3D while, by adding…

A: Given: It is given that, the value of P and Q is as follows: ...... (1) and ......(2) To prove: we…