12.55. Prove the following: Let a, b, m, n e Z, where m, n 2 2. If a = b (mod m) and a = b (mod n), where gcd(m, n) = 1, then a = b (mod mn).

Q: 10. Prove that if o is the m – cycle (a,a2 ... am), then for all i e {1,2, ...m}, o'(ar) = ar+i,…

A: Given that σ is the m- cycle (a1,a2,...am). We need to prove (i) σi(ak)=ak+i mod m for all i ∈{1, 2,…

Q: Let n E Z+, n > 2. Show that n is prime if and only if (n – 2)! = 1 (mod n).

A:

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

A:

Q: Let x, y, z E Z and x = y mod n, where n ez+ Then: 1) Show that x - z = y -z mod n 2) Show that xz =…

A: The given question is about number theory. The solution is given below.

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: 8. Find the remainder of x2021 + x2020 + 1 modulo x2 – 1 in Q[x].

A: Remainder of the Polynomial

Q: 4,75. Let n, m E Z. Prove that if n = 1 (mod 2) and m = 3 (mod 4), then n + m = 0 (mod 4).

A:

Q: For integer n>1, nZ={nx:x in z}, nZ is set of all integer multiples of n. Prove that integer p is…

A:

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

A:

Q: 1. Let a, b e Z. Prove that if a = 0 (mod 5) and b = 2 (mod 5), then a² + b² = 4 (mod 5).

A:

Q: 13. Compute 41023 (mod 11). a. 14. Prove that for any integer n, n31 = n(mod7).

A:

Q: Prove, by induction, that, for n2 5, the nth Fibonacci number satisfies Fn s 2".

A:

Q: 15 Composite Then and O modn Show that if n there are with X#0 od n and y7 but where yy= Omod n.

A: Given n is a composite number

Q: 13. (a) Prove that for each integer a, if a 0 (mod 7), then a #0 (mod 7).

A:

Q: - Prove that for all n E Z, we have ged(27n – 20, 4n – 3) = 1.

A: Solution

Q: 1. In the proof of the claim "there is no integer n such that n = 2 mod 6 and n = 7 mod 9," the…

A: See answer below

Q: For which positive values of k is nk = n (mod 7) for all integers n?

A:

Q: Consider the pseudorandom number generator given in our lessons: with Uf +1 = (auf + b) mod m…

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Q: 9. Find all integers x, 0 < x < n, satisfying each of the fol- lowing congruences mod n. If no such…

A: Q (d),(g) asked and answered.

Q: Let C={x €Z: x =7 mod 9} and let D={x € Z,x = 1 mod 3). a) List five elements of C and five elements…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: For which p= 3 mod 4 is i a square in F»[i]? For which p = 3 mod 4 is i the square of an element in…

A:

Q: Let n be a positive integer. We write x; for the ith digit of n. (I.e., if n = 217, then x1 = 7, x2…

A:

Q: () 2p n! (a) (") = 2 (mod p), where k for 0 <k <n. k!(n – k)! (b) (Зр - 1)! 2р2 (mod p3)

A: As per our guidelines we are supposed to answer only fast question. Kindly repost other part in the…

Q: 2. Let A = {n? +n € Z:n e Z} and B = {r e Z : r³ – 3x³ + 2x = %3D O mod 5}. Prove or disprove: A C…

A:

Q: (a) Prove that for all n 2 0, Eo fi = fn+2-1 where fi-1 is the th Fibonacci number.

A: Fibonacci sequence

Q: 7. Let n E Z. Prove that 9n3 6 (mod 27).

A: The objective is to prove the above statement.

Q: 7. Let p and q be primes and suppose that p = 2q + 1. Let h E F and assume that h 1. Pre that ² h…

A:

Q: Show that th equation x" + y² = z" (mod 5) has nonzero solutions in Zz if and only if n is odd.

A: Given: xn+y2n≡zn(mod 5)

Q: Consider the subring S = 2216 of Z16. Then char (S) = 4 6. 8. оОО О

A:

Q: 20. compute : 11 (21018) * for number 17-20 Let o:2x2-2 be defined by a 06= (a-b) mod 37.

A: We will use the given definition of the operation.

Q: Let a be an integer and let n e N. (a) Prove that if a = 0 (mod n), then n | a.

A: By the definition if congruence let a, b and n are integer, with n &gt;0, then a is congruent to b…

Q: 4.14. Let a, b, n E Z, where n > 2. Prove that if a = b (mod n), then a2 = b (mod n). 4.15. Let a,…

A: The detailed solution is as follows below:

Q: 3) If the gcd(z,n) = d and xz = yz mod n then: Show that x = y mod

A:

Q: Let n E Z. Prove that 3n # 1 (mod 9).

A: Proof by contradiction: Direct approaches can be difficult (or impossible) to use to prove a…

Q: 1) Using the encryption function f(x) = (10– x) mod 26, 0sx s25, to decrypt the message "DKPG K XCIG…

A:

Q: 4) The least positive integer residue of 21000000 mod 17 is a) - 1 b) 1 c) 2 d) a

A:

Q: For D = {1, 2, 3, 4, ..., 999}, find a counterexample to the claim, "For all n e D, if n mod 3 =1…

A: Given D=1, 2, 3, 4, ..., 999 Here, we have to find a counterexample to the claim,  ∀n∈D, if n mod…

Q: 12. Let the "Fibonacci-2" numbers gn be defined as follows: 91 = 2, g2 = 2, and gn+2 = (gn+1)( In)…

A:

Q: 2. (a) If gcd(a , 35) = 1, show that a2 = 1 (mod 35). [Hint: From Fermat's theorem a = 1 (mod 7) and…

A: Fermat's Little Theorem:- It states that if p is any prime and a is any integer then, ap≡amodp…

Q: For n=195, by Fermat's theorem d-2 a) a (modn) b) 1(modn) c) 0(mod n) d) a(mod n)

A: Given:                 n=195 To find:                an-2=? by using Fermat's theorem Fermat's…

Q: For D = {1, 2, 3, 4, ..., 99}, find a counterexample to the claim, "For all æ, y € D, if æ mod 3 = y…

A:

Q: Prove that whenever x ∈Zand d ∈Z+, if d does not divide (x2 −3x), then x mod d does not divide 3

A: The given statement is False.  Consider, x = 5 and d = 4. Then we see, x2 - 3x = 52 - 3x5…

Q: 1 Prove each of the following assertions: (a If a = b (mod n) and m |n, then a = b (mod m). (6) If a…

A: We know that  a ≡ b (mod n) is equivalent to n | (b-a)

Q: Q3. Prove that if kx = ky (mod m) , then x = y (mod m/d) where d = gcd( m, k).

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

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Q: 4.18. Let m, ń EN such that m|n. Prove that if a and b are integers such that a = b (mod n), then a…

A: The detailed solution is as follows below:

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…

A:

Q: For any integers m and n, if m mod 11 is 6 and n mod 11 is 7, what is m+n mod 11? 13 9 42

A:

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.