Let a, n E Z+, where gcd(a, n) = 1. Show that if n is a pseudoprime to the base a, then n is a pseudoprime to the base ā, where ā is an inverse of a modulo n.

Q: Find each of the following, fully justify your claims. (a) Suppose that n div m = 2021 and that n…

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Q: Recall that the Fibonacci numbers are defined as follows: Fo = 0, F1 = 1, for all integers n> 1, F,…

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Q: Use strong induction to show that when n> 3, fn> a"-2 where fn is a Fibonacci number and b. a= (1+ v…

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Q: • For all integers n and d such that d> 0, n mod d= r + (r e Z, 0<r< d, and) n = dq + r for some…

A: Question: Prove that For all integers a, if a mod 3=1 then a3=a-13

Q: and drag on elements in order hese statements in order to produce a proof that the integers a and b…

A: Introduction: If the difference between integers a and b is a multiple of n, we say they are…

Q: Give an inductive proof that the Fibonacci numbers Fn and Fn+1 are relatively prime for all n ≥ 0.…

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Q: Let p be a odd prime number and b, c € Z. Assume that there exists l e Z, such that l² = b² – 4c mod…

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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: 1. Let n = pq, where p and q are distinct odd primes. Define X(n)= (p-1)(q-1) gcd(p-1,q-1)* Suppose…

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Q: Let n be a positive integer, and a an integer Prove that if the order of a modulon is n-1 then n is…

A: The given problem is to prove that n is prime from the given problem given a modulo n is set of…

Q: The number p = 65537 = 210 +1 is known to be prime; you may take this as given. Let g be a primitive…

A: Since, g is a primitive root modulo p, order of g is (p-1) = 216 Now, we know order of gk is…

Q: Find an integer a with 572 < a < 1144 such that a is a solution to the following system of…

A: This question is about application of Chinese remainder theorem

Q: . Show that Z with addition modulo m, where m≥ 2 is an integer, satisfies the closure, associative,…

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Q: Let p be a prime number. The polynomial p(x) = x²+x+1 is irreducible over Zp if and and only if…

A: Given that for a prime number p, the polynomial px = x2+x+1 is irreducible over ℤp if and only if…

Q: Let m e N and a E Z. Suppose that 3|(m) and that a is a primitive root (mod m). Prove that a is not…

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Q: Suppose that p is a prime such that p = 5 (mod 8). Let k E Z such that p = 8k+5. For any a E Zp,…

A: 1. Let a∈ℝn. Then a is called quadratic residue modulo n if there exists an integer x such that…

Q: Suppose that an is defined by setting a = = -2, a₂ = 0, and an = 4an-1-4an-29 where n ≥ 3. Using…

A: Since you have posted multiple questions, we will solve the first question for you. If you want any…

Q: If dįd2dzdądzded7ds is 1 4 0 5 7 3 2 3, let n = 100 + dzds and a = dzdg. n 3D Find d e Z, such that…

A: The solution follows from the Generalized Euclidean Algorithm for finding GCD.

Q: Let a, n be positive integers. Knowing that 15a - 13n = 1, we deduce that the multiplicative inverse…

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Q: For every number a e a e {1, 2, 3, ..., 18} which is not a primitive root (modulo 19) find the…

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Q: Let n E Z', where n > I. Show that if n is a product of distinct Mersenne primes, then o(n) = 2k for…

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Q: Let a and b be positive integers with (a, b) = 1. Show that if g is a primitive root (mod ab), then…

A: We have given (a,b)=1 where a, b both are positive integer. We have to show if g is primitive root…

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

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Q: DuppOse that a anа т аге intege Such that a, аre al diпerent IMod- ulo m, with am-1 = 1 mod m. Show…

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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: Find s e Z with 0ss< 13, so that s = 127+300(mod 13). Your work should show a method that does not…

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Q: Let p be prime. (a) If a 1 (mod p), show that 1+ a + ...+ aP-1 = 1 (mod p). (b) If a -1 (mod p),…

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Q: Use Fermat's theorem, show that if gcd(a, 145) = 1 then a141 = a (mod 145). !! imun m nunY

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Q: Let n 2 1, x be a real number, and r > -1. Prove the following statement using mathe- matical…

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Q: Let n E Z. Prove that n3 – 7n + 18 is divisible by 6 for every interger n. Do not use induction.

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Q: Define the following operation on Zn: S [1] if a = b (mod 5) (a] + [b] = { joj if a ±b (mod 5) %3D…

A: Consider the provided question, The provided operation on Zn is given by, a*b=1    if a≡b(mod5)0…

Q: Show that the Bernoulli number B₁ satisfies B₁ = (-1)" ( 2 + 2d²+1) Σ (mod 1) for any positive…

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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: Let C(n) be the constant term in the expansion of (x+5)". Prove by induction that C(n) = 5", for all…

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Q: Let p be an odd prime and r a primitive root of p. Show that if a and b are positive integers in Zp,…

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Q: consider the following table Classes |Frequency 25-29 30-34 15-19 1 20-24 35-39 12 10 10 Find the…

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Q: Let min EN 7 atR he such that ca,m)=1, Ca,n)=l ģ (min) =L, Let K= Om(a) } j=OnCa). Suppore Prove…

A: Let m, n∈ℕ and a∈ℤ be such that a, m=1, a, n=1 and m, n=1. Let k=oma and j=ona. Suppose j | k, prove…

Q: Let each of a and m be an integer with m > 1. We know that if a and m are relatively prime, then a…

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Q: Find an integer a with 572 <a < 1144 such that a is a solution to the following system of…

A: This question is about application of Chinese remainder theorem

Q: Let p be an odd prime and a ∈ (Z/pZ)x be such that (a/p) = 1. Show that a is a square modulo every…

A: Given: Let p denotes an odd prime and a ∈ (Z/pZ)x be such that (a/p) = 1.

Q: If a is a primitive root modulo n, then a¯1 is a primitive root modulo n O False True

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Q: Consider the se quence defined recursively as follows do =2 ak =yak-itl © find di,a2,93,a4 ,as ®…

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Q: Every natural number n has a base-60 expansion; that is, there exists k > 0, and for each i = 0, 1,…

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Q: 5. If a-l is the inverse of a modulo m and b-1 the inverse of b modulo m, show that a--1 is the…

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Q: . Let b, k and m be positive integers that satisfy gcd(b, m) = 1 and gcd(k, ø(m)) = 1, %3D where…

A: Here GCD(b,m)=1 means they are coprime.

Q: Consider the relations E9 and E12 on the set of integers defined as follows • aE9b precisely when…

A: Given relations aE9b precisely when a≡bmod 9 aE12b precisely when a≡bmod 12 where relation R=E9oE12.

Q: In parts (a)- (e), involve the theorems of Fermat, Euler, Wilson, and the Euler Phi-function. (a)…

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Q: Let n > 2 be a positive integer and r a primitive root of n. Suppose that a E Z is such that ar? 1…

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