Let a, b e Z and n E Z*. Prove that if a = b(mod n), then gcd(a, n) = gcd(b, n).

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

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Q: e) Define W: Z -Z and V: Z → Z by the rules W(a) = a² and V(a) = a mod 5 for all integers a. Find…

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Q: Suppose gcd(a, n) = 1. Show that there exists b such that [a]n * [b]n = [1]n.

A: Using division algorithm.

Q: Let a, b be integers. What does it mean for a = b (mod 17)? Describe the elements of Z17. Compute…

A: a ≡ b (mod 17) means that a − b = 17k for some k ∈ ℤ. ∴  b − a = −17k = 17(−k); henceb ≡ a (mod…

Q: Show that, for given integers a, b, c, if gcd(a, c) = gcd (b, c) = 1, then gcd (ab, c) = 1

A: Given a,b,c∈Z+ And gcda,c=1 and gcdb,c=1 We have to show that gcdab,c=1 Since, gcda,c=1 and gcdb,c=1…

Q: Let a, b, n be positive integers. Prove that if a = b (mod n), then GCD(a, n) = GCD(b,n).

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

A: Use the following definitions, to obtain the solution. The Euler phi function for the positive…

Q: Let A = {r € Z | r mod 15 = 5} and B = {x € Z| r mod 3 = 2}. Give an outline of a proof that A C B,…

A: Suppose we have two non-empty sets A and B . If x∈A then x∈B for all x∈A then A⊆B . If x∈B then x∈A…

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: 1. Prove that if ā and be Zn are invertible modulo n, so is ā - b. Is this the case for ā + b?…

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Q: Prove that if r <s < n, then P(n , s) is divisible by P(n,r).

A: Given: r≤s≤n Therefore, n-r≥n-s So we can write n-r! = n-rn-r-1n-r-2...n-sn-s-1n-s-2...1 n-r! =…

Q: The linear congruence ax = b(mod m) has a unique solutionif and only if: gcd (a,m) =1 gcd(a,m) =1 o…

A: First option is correct. gcd(a, m)=1.

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

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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: 6. Let a € Z and assume that gcd(a, 55) = 1. Prove that a40= 1 (mod 55). (Hint: Fermat's little…

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Q: Consider the set Z, × Z, = {(x, y) : x € Z, and y € Z3}. Define the operation/sum O in Z2 × Z3 such…

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Q: 2. Let a be a positive integer. Explain why the linear congruence az b (mod a + 1) has a solution in…

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Q: Prove that pork = pur in O₂ if and only if 0 w = 2m for some m€ Z and k = 1 (mod 2)

A: To prove that ρθrk=ρωrl in Ο2 if and only if θ-ω=2πm for some m∈ℤ  and k¯≡ l¯mod 2.

Q: Let d, t € N be such that d | t. Prove that if x, y € Z are such that x = y (mod t), then x = y(mod…

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Q: For each divisor k>1 of n, let U,(n) = {x E U(n) | x mod k = 1}. [For example, U,(21) = {1,4, 10,…

A: For each divisor k &gt; 1 of n , Uk(n) =x∈U(n)| xmod k =1. To Find: List the elements of U420,U520,…

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: The linear congruence ax = b(mod m) has a unique solutionif and onlyif: @cd(a,b) = 1 cd(a,m) =1…

A: Second option is correct.

Q: Let a , b >= 1 and x ∈ Z. Prove that if xb ≡ 1 (mod a) then gcd(x, a) = 1

A: Let a , b ≥ and x ∈ Z  Suppose xb ≡ 1 (mod a)  ⇒a divides xb-1  By defination of divisibility, there…

Q: Define L:Z → Z and M:Z → Z by the rules L(a) = a², M (a) = a mod 5, V a ɛ Z. Find the values of (a)…

A: Composition of two functions f(x) , g(x)  is defined as : (f∘g)(x)=f(g(x))         If we have to…

Q: For which positive integers a is the congru

A: Given,               ax4≡2 (mod 13)

Q: The greatest common divisor c, of a and b, denoted as c = gcd(a,b), is the largest number that…

A: Here is the solution

Q: 3. For the following statements, either prove that they are true or provide a counterexample: (a)…

A: Given that, Let a, b, m, n ∈ Z such that m, n &gt; 1 and n | m. a ≡ b (mod m) By using the…

Q: (2) Show that if a, b € Z/nZ has multiplicative orders k and I respectively and (k, l) = 1, then ab…

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

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Q: c. Let o, = (2 5 3 6) and o, = (1 4 6 5 7) be two permutations of S,. Find (0,02)¯1.

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Q: For each divisor k . 1 of n, let Uk(n) = {x ∈ U(n) | x mod k = 1}. [For example, U3(21) = {1, 4, 10,…

A: For each divisor k &gt; 1 of n , Uk(n) =x∈U(n)| xmod k =1. To prove Uk(n) is subgroup of U(n).

Q: Let i, r, and k be positive integers, and let s gcd(i, r). Assume that r | ik. Prove that | k.

A: To prove the divisibility property under the given conditions

Q: Let x, y, u, v € Z and d E N. Prove that if x =y (mod d) and u = v(mod d), then xu = yv (mod d).

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