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

Q: What sequence of pseudorandom numbers is generated using the linear congruential generator xn+1 =…

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Q: show that for all integers x and y (x-y)3=x3-y3(mod3)

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Q: Find the smallest positive integer solution to the following system of congruence: x≡17(mod35)…

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Q: 2=1 or 5 (mod 6) , and if there exists a A-fold triple system of v, prove that v = 1,3 (mod6)|

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Q: Show that x2 = Y mod p has

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Q: If A ánd B are two n x n invertible matri hen one of the following is always correct

A: It is given that,  A and B are two n×n invertible matrices.

Q: 1. Prove that the followingn is a theorem in SD. Av - A

A: Theorem :-

Q: If gcd(a, n) = 1, then the congruence ax = b modn has the solution x = - ba®(n)-1

A: Given that gcd(a,n)=1. We need to check whether ax≡b mod n has the solution x=ba∅(n)-1

Q: Now let ap be a nonzero element in Z/pZ. Show that a, is invertible. (Hint: Invoking the fact that a…

A: Let [a]p be a nonzero element in Z/pZ.

Q: 2=1 or 5 (mod 6), and if there exists a A-fold triple system ofv, v= 1,3 (mod6)| prove that

A: ANSWER:- Given that, λ=1 or 5(mod 6)   = 1 mod(6) or 5 mod(6) λ=1*6 or 5*6  =6 or 30

Q: 2.6. In the text, the modulus is defined in terms of the inner product. Prove at this can be turned…

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Q: Recall that tn denotes the nth triangular number. Prove the following identity: tn+2k = tn (mod k)

A: If tn is the nth triangular number, then tn=nn+12

Q: Find the sequence of pseudorandom numbers generated by the linear congruential method with modulus m…

A: The generator is defined by the recurrence relation:

Q: 2. Prove that L(cashkt)= ki 52-K2

A: To prove that  L(cosh(kt))=ss2-k2

Q: Let [a] ϵ Zn- Prove ([a]) = ([n - a]).

A: Given, a∈Zn- The objective is to show that, a=n-a. Let x∈a. Then, x=ra,r∈Zn=ra=-rn-a=-rn+ra as…

Q: 2. Prove that if 5BA+ AC A - I = A, then A must be inver

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Q: State the third congruence that must be given to prove that AABC = ADEF using the indicated…

A: 1. For the ∆ABC and ∆DEF, it is given that ∠B≅∠E and BC≅EF. To prove ∆ABC and ∆DEF congruent using…

Q: Find all values of that are solutions to the following system of congruences (Show your work!): x =…

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Q: 2. Let R = {(x, y) : x, y E R}. Prove or disprove that R is reflexive, symmetric and/or transitive…

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Q: Prove that gcd(a, b) is the least natural number representable in the form ax + by with x, y ∈ Z.

A: We have to show that gcd(a, b) is the least natural number representable in the form ax + by with x,…

Q: 2. Prove that if 5BA+ ACA – I = A, then A must be invertible.

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Q: Prove 2- = =4 has infinite many Solu thans ZEC.

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Q: 1 6 [: 2. Let %3D matnix c (by tnial and erron) Constauct a C (by tnial and eror using anly !,-1 and…

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Q: Use Hensel's Lemma to find a solution to the following congruence: a +3 x + 3 = 0 mod 54.

A: Let f(x) be a polynomial , p is a prime number . If x=a is a solution of congruence equation…

Q: 2X+3=1 mod 5 Show your work

A: We have solve given problem:

Q: Use Fermat's theorem, show that if gcd(a, 155) = 1 then a121 = a (mod 155). %3D

A:

Q: 10. Prove that Z has no smallest element.

A: we will prove it by contradiction, Suppose x belongs to Z  is the smallest element.

Q: Use the construction in the proof of the Chinese remainder theorem to find all solutions to the…

A: To solve the system of congruences: x ≡ 2( mod 3)  x ≡ 1( mod 4) x ≡ 3(mod5)

Q: ar – 5ak-1+ 6ak-2 = 5k, ao = 1, az = 3

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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: Find the first five terms of the séquence an= 1 as as express in Standard form

A: Given  an = 1+ 1/n  To find the first five terms

Q: Suppose a community of RSA users share the same encryption exponent e but each user i has their own…

A: Show that if k>e, then a cryptanalyst can recover the plaintext x without factoring any of the…

Q: Find the smallest positive integer satisfying the system of congruences: x ≡ 1 (mod 3) x ≡ 2 (mod…

A: Here we will find out the required value.

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: 5. In each Part determine whether the matrix is invertible modulo 26. If so, find its inverse modulo…

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Q: (b) Consider the system of linear congruences below: x = 2 (mod 5) 2x = 22 (mod 8) 3x = 12 (mod 21)…

A: As per bartleby guidelines, for more than one questions asked, onlyone is to be answered, kindly…

Q: What sequence of pseudorandom numbers is generated using the linear congruential generator xn+1…

A: Determine the sequence of pseudorandom numbers generated using the linear congruential generator…

Q: Asbulbr 5t2om is lo ls tals hagabnsiedmannsvs 4. LetA = {0,1,2,3,4} and f: A → A where f(n) = 3n mod…

A: If a and b are integers and n > 0, we write a ≡ b mod n to mean n|(b − a). We read this as “a is…

Q: Solve the system of congruence's x = 2 (mod 3), x = 4 (mod 5), x = 1 (mod 7)

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Q: Q3/ Use Chinese theorem to solve the following congruence: X=2 mod 3 X=1 mod 5 X 2 mod 4

A: Since you have posted a multiple question ,I will solve the first question(i.e.Q3)  for you. To get…

Q: Find all solutions of quadratic congruence x^2+85x+2=0(mod401)

A: We follow the method of completing the squares of the equation x^2+85x+2. Note that 85 is odd so we…

Q: Find an inverse for the linear congruence 9x = 15 (mod 23) by using the Euclidean algorithm to…

A: The given problem is to solve the given linear congruence equation by finding the inverse using…

Q: Use the given information to state the third congruence that is needed to prove AMN0=A RST by the…

A: The AAS rule states If two angles and a non-included side of one triangle are equal to two angles…

Q: 1. Find all solutions to the system of congruence. (Use CRT) x = 1 (mod 2) x = 2 (mod 3) x = 3 (mod…

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Q: Find the integer a, if the modular representation of a is (2, 1, 5) with respect to the modulo 3,…

A: We have to find the integer a if the modular representation of a is 2,1,5 with respect to modulo 3,…

Q: Exercise 25.2. Prove that log, (bc) = log, b+ log, c (mod p- 1) for all b, c not divisible by p.

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Q: a) Find the sequence of pseudorandom numbers generated by the linear congruential method with…

A: Given that, the equation is xn+1=axn+c mod m and the values are m=9,a=65,c=2 and x0=5. Obtain the…

Q: Show that on finite m

A: To prove that on finite measure, Lpand almost everywhere convergence are independent.

Q: Find the positive integers n such that , the sum of positive divisor oln) = 24,

A: Sum of Divisors: Let the prime factorization of n be n=p1a1×p2a2×...×pkak Then the sum of divisors…

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.