(3). (a). Find all solutions to the congruence 6x = 9 (mod 45)

Q: The sum of any two integers of the from 4k + 1is always: Od Zero Even prime

A: Sum of two integers of the form 4k+1

Q: 2. Investigate the definiteness of the following quadr

A: Definiteness of quadratic form: It might be positive definite, negative definite, or indeterminate…

Q: Find the z tronsfom of f(n) = a" cos (bn).

A: We have to find the z transform of the following.

Q: (b) Prove by induction that (3i – 2)² = n(6n² – 3n – 1) for n 21 %3D i=1

A: Prove the equation ∑i=1n3i-22=12n6n2-3n-1 for all n≥1. Prove the equation for n=1. Substitute n=1…

Q: 2. Solve the following polynomial congruences. ces (a) x² + 12x – 17 = 0 (mod 143) (b) x³ + 30x² +…

A:

Q: The equation 82r = 12 (mod 13) has O 1 incongruent solution modulo 12 O 4 incongruent solutions…

A:

Q: 3.1.4 Find all integers x which satisfy the congruence equation 3x = 2 mod 8.

A: 3x = 2 (mod 8) So, 3•(3x) = 3•2 (mod 8) So, 9x = 6 (mod 8) So, x + 8x = 6 (mod 8)

Q: *Find all solutions of each of the congruences: x2 + x +1 = 0(mod11) (a)

A: To find the solutions of the polynomial congruences: a) x2+x+1≡0 mod 11 Let f(x)=x2+x+1 x 0 1 2 3…

Q: and C of S such that oB oc (mod 6). 5.50. Prove that the equation cos (x) – 4x + 7 = 0 has a real…

A:

Q: Prove that 2" < n! for any integern 2 4

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Express in the form & j(n/2-j) (2) In(3

A: As per the bartleby guidelines for more than one part only one has to be solved. Please upload the…

Q: (c) Find a value t modulo 7 so that (3+ 7t) does not solve the congruence x® – 1 = 0 (mod 49).

A:

Q: Find a closed formula for a, if ak = 5n² + 4n k=1 an =

A:

Q: then x* 1 is odd." Give a proof by contradicton to show "1fx s even integer,

A:

Q: {-74 + 1, ..., 74 – 1}. (b) Hence, find a solution to the congruence x' = 99 mod 74.

A: Given:- An expansion above Polynomial will have co-efficient lies between {-74+1,..........,74+1}

Q: 1) Find a solution – if possible – for each of the following two congruences. If for one of the…

A:

Q: Find all solutions to the following quadratic congruences by first making the leading coefficient 1…

A: Topic:- algebra

Q: 5. Prove that the equation ø(n) = ¢(n+ 2) is satisfied by n = 2(2p – 1) whenever p and 2p – 1 are…

A: Euler totient function formula: p is prime then ϕ(p)=p-1 ϕ(pk)=pk-pk-1     ∀k∈ℕ   If gcd(a,b)=1…

Q: 1. Find all x € ZZ satisfying each of the following equations. [from # 1, 3.5] a) 5x + 1 = 13 (mod…

A:

Q: nd to wnwyun g ans 2

A:

Q: 4. Find the modulus and argument of z = -2-2i"

A: Modulus of a Complex Number: Let z = x + iy is a complex number. A complex number represents a point…

Q: (2) (a) Find all of the solutions modulo 15 to 6x = 15 mod 18. Express your answer in terms of the…

A: Given, 6x≡15 mod 18. As it is known that, an≡bn mod cn ⇔ a≡b mod c. Therefore, we have 6x≡15 mod 18…

Q: 7. Find a number t such that In(4t +3) = 5 (Give EXACT value)

A: Given that ln(4t+3)=5 we need to find "t"

Q: 1. Find all solutions to the following congruences. (a) 2x +3 = 7 (mod 9) (b) 2x +3 = 7 (mod 20) (c)…

A:

Q: 1. Solve the following linear congruences using Diophantine equations: 25x = 7 (mod 17) and 26x = 8…

A: Given information: The Diophantine equations are 25x≡7mod17 and 26x≡8mod6. Calculation: The equation…

Q: Theorem 6 If (a1 + a3 + a5) > (a2 + a4) and (B1 + B3 + B5) > (B2 + B4), then the necessary and…

A: Given that ym+1=Aym+α1ym-1+α2ym-2+α3ym-3+α4ym-4+α5ym-5β1ym-1+β2ym-2+β3ym-3+β4ym-4+β5ym-5…

Q: Theorem 6 If (a1 + a3 + a5) > (a2 + a4) and (B1 + B3 + B5) > (B2 + B4) , then the necessary and…

A:

Q: Q1: Prove each of the following: Z1 + z2 = Z1 + Z2 Z1 z2 = Zī Z2 21 图= 1. 2. %3D 3. Q2: Convert to x…

A: Find the attachment

Q: 5. Solve the following quadratic congruence using the "completing the square" method: 9x² +66x +49 =…

A: 5. Given equation is 9x2+66x+49≡0(mod 79)   To Solve: Above quadratic congruence using the…

Q: 8. Solve the following quadratic equation via the two different methods, namely both classical and…

A:

Q: 1. Prove the following using direct proof. a. If 5x – 3 = x + 5, then x = 2. b. If x and y are both…

A:

Q: Using Fermat's little theorem, find the solutions of the following linear congruences. a) 7x ≡ 12…

A: Fermat's theorem :  Let p be prime number such that gcd(a,p)=1 then ap-1≡1(mod p) Using Fermat's…

Q: 13. Solve the rccursive formula an = an-1 +1+2"-1 with ao = 0. Verify by %3D induction.

A: T

Q: 9. Determine which of the following congruences have solutions, and find any solutions (between 0…

A:

Q: (b) Prove by Induction that (3i – 1)(3i +2) = 3n +6n2 +n for n21 i=1

A:

Q: 1. Solve the following linear congruences: (a) 25x = 15 (mod 29). (b) 5x = 2 (mod 26). © 6x = 15…

A: As per our guidelines we can answer only three subparts and rest can be reposted.  So, I am going to…

Q: B. Using Fermat's Theorem •Determine all solutions of the equation 6x = 18 in Z24. •Describe all…

A:

Q: 1. Prove that if 7n2 + 5 is even, then n is odd

A: The given statement is,

Q: ) ) - 1 3 1 3. (a) Without using a calculator, prove that 2 2n +1 Зп + 1 (b) Prove that for all…

A: 3(b) Prove the identity for n = 2. For…

Q: 3.2.6 Find all solutions x to the congruence equation 12x = 1 (mod 17)

A: Given, 12x = 1 (mod 17) So, 7•12x = 7•1 (mod 17) So, 84x = 7 (mod 17)

Q: Express each of the following in the form a + bi. 4. cosh(1-12)

A:

Q: solution to the congruence equation (3x + 12) = 7 mod 10

A: Given :  3x+12=7mod 10⇒     3x=7-12 mod 10⇒     3x=-5 mod 10⇒     3x=5 mod 10        -5 mod 10 =5⇒…

Q: 1) Guess a formula for the sum 1-3+5-7+..+ (-1)*-(2n – 1) =(-1)-(2i – 1) i=1 and prove your fornmula…

A: The given sum is  1-3+5-7+...+-1n-12n-1=∑i=1n-1i-12i-1

Q: Determine a general congruence condition satisfied by all primes p with p = 3(mod4) such that the…

A: As per the question we are given a number theory problem. Where we have to find a congruence…

Q: 2n + 1 6n+3 Show, by using congruence, that 11 divides 3 +2 for n21

A:

Q: 1. Find all solutions to the following congruences. (d) 5x = 10 (mod 20)

A:

Q: 2. Find all solutions of each of the following linear congruences. c) 17x = 14 (mod 21) a) 3x = 2…

A: Given: To find the linear congruence of d)15x≡ 9(mod 25)e)128x≡833(mod 1001)f)987x≡610(mod 1597)

Q: 10. Solve the following quadratic equation via the two different methods namely both classical and…

A:

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: