Prove that if gcd(a, m)=1 and gcd(a - 1, m) =1, then 1+a+a²+...+ a² + a(m)−1 = 0 (mod m).

Q: Find the first three values of the linear congruential generator X; = 2x;–1 +3 (mod 13) starting…

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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 (X, E) be a measurable space. Let py and pz be two measures defined on X. Then 1. p - H2 is a…

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Q: Let a, b, c E N. (a) Prove that if gcd(a, c) = 1 and gcd(b, c) = 1, then gcd(ab, c) = 1. (b) Prove…

A: a, b, c∈ℕ

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

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Q: Consider the linear congruence ax = b (mod m). If (a,m) = 1, then this has a unique solution x = a¹b…

A: Since gcd(a,m)=1. ad+mk=1 for some integers d and k. ⇒ad=1-mk⇒ad=1 (mod m) So a is invertible (mod…

Q: = Let An= a₁ + a2 + ... + an, Bn=b₁ + b₂ + b3 + +bmDn C + C₂+...+c, and cn=a1bn + a₂bn-1+...+ab₁ne…

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Q: Let Z3[i] = {a + bi : a, b ∈ Z3, i2 = −1} where arithmetic is done mod 3. 1. What is |Z3[i]|? Give…

A: Given , Z3[i] = { a+bi : a , b ∈ Z3 , i2 = -1 } (.)  Zp[i] is a field if and only if p is a prime…

Q: Prove that gcd(a, b) divides gcd(a, bc). Prove that if gcd(a, b) = 1, then gcd(a, bc) = gcd(a, c).

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Q: Show that if m e Z+ and a e Z such that gcd(a, m) = 1 and ordma = m – 1, then m is prime.

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Q: left cosets

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Q: x and y are two positive integers such that y > 5x. Select the expressions that are not equal to…

A: We choose x and y such that we are able to discard options.

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: Calculate (n | ² | m) and (m | 4 | n) in the N-representation; | n) and | m) are harmonic oscillator…

A: As per the question we are given two distinct state vectors |n⟩ & |m⟩ (harmonic oscillator…

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

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Q: Suppose that a divides b and that c divides b. Suppose that gcd(a, c) = 1. Prove that ac divides b.

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Q: 3. Let h: Z→ Z be the function h(r) = = x mod 4. Find h-¹(0), h-¹({0, 1, 2, 3}), and h-¹(4). Explain…

A: In this question we can easily observe that that output of the function is the remainder when any…

Q: Prove that ||A||1 = max1<j<n{£", |aij|}. %3D

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Q: Prove that gcd(k; n) = gcd(n - k,n).

A: Greatest common divisor (gcd) :            If a , b are two integers such that atleast one different…

Q: Assuming that gcd(a,b)=1, prove the following (a)  gcd(a+ b,a - b)=1 or 2. [Hint: Let d=gcd(a+ b,a…

A: Let d=gcd(a+ b,a - b)    =&gt; d | (a + b) and d | (a - b)     =&gt; d | {(a + b) + (a - b)} and d |…

Q: Find all gcd’s of the two Gaussian integers 5 + 3i and −5 + i.

A: By repeated use of Euclidean division algorithm  we get the value of g.c.d.

Q: give a combinatorial proof of the identity  P(n,k) * (n-kCm-k) = (nCm) * P(m,k) where all…

A: Detailed proof is provided in the next step.

Q: Show that gcd(a,n)=1 if these exist m and k such that ma+nk = 1.

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Q: Compute the following (over Z2[x])(a) x3(x2 + x + 1) (mod x3 − 1).(b) (x2 + 1)2 (mod x3 − 1)

A: Step 1:To compute these expressions over Z2[x], we need to perform polynomial arithmetic under…

Q: Let a and b be integers with GCD(a, b) = d and let a = da', b = db'. Prove that GCD(a', b')=1

A: according to guidelines we can do only first question thank youLet a and b be integers with GCD(a,…

Q: Are the polynomials x³+2x +2 and x4 + 3x -2 equal to mod(x² + 2x + 2) in Q[x]?

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Q: (a) Solve the following simultaneous system of congruences : x = 2(mod 4) x = 1(mod 5) x = 8(mod 11)…

A: (5) (a) x≡2(mod 4)x≡1(mod 5)x≡8(mod 11)As  g.c.d.(4,5,11)=1 , hence the symaltaneous solution exists…

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: Let a, b, c, and m be positive integers. (a) Prove that the linear congruence ax = c (mod m) has…

A: Solution: a) Consider the congruence  ax≡cmod m---(1) Let, the congruence equation (1) has a…

Q: Determine if x¹ + x² + x + 1 is irreducible in Z[x].

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