2. Solve the following polynomial congruences. ces (a) x² + 12x – 17 = 0 (mod 143) (b) x³ + 30x² + 27x + 23 = 0 (mod 45)

Q: Solve each of the following congruences. (a) 6x + 3 = 4 mod 10 (b) 6x + 3 = 1 mod 10 (c) x² = 1 mod…

A: In congurent modulo n, ax≡bmod n has exactly 'd' solutions if 'd' divide 'b'where d=gcda,n otherwise…

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: Either solve the congruence, giving the solution as a least residue, or show that it has no…

A:

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: If p times tne pth term of an A.P. is equal to q times the qth term of the same AP., then how that…

A: Given that If p times the pth term of an A. P. is equal to q times the qth term of same A. P.  To…

Q: Suppose f(x) is a degree-2 polynomial. How many solutions x are there to f(x) = 0 (mod 37 · 59) such…

A:

Q: The following recurrence relation is non-homogenous. Solve: an - 5an-1 + 6an-2 = n with a0 = 0 and…

A: Solve: an - 5an-1 + 6an-2 = n with a0 = 0 and a1 = 2

Q: Find the solutions for the following recurrence an 9an-1-18an-2, with initial conditions %3D dо 3D…

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

Q: (b) Since 61 is a prime with 61 = 1 (mod 12) the polynomial X? + 3 has roots in Z/61Z by (a): they…

A: Here we need to solving ....(i) knowing that .....(ii)

Q: 3. Find integers s and t so that 1 7s + 11t. Show that they are not unique.

A: The division algorithm states that for any integer, a, and any positive integer, b, there exists…

Q: Find a number t such that In(4t + 3) = 5 %3D

A:

Q: solve the quadratic congruence 7x^2 - 8x +4 = 0 (mod 17)

A: The given problem is to solve the given quadratic congruence equation 7x2-8x+4 = 0(mod 17)

Q: (d) Find an integer x, 0 < < 18, that satisfies the following congruence: 9x + 13 = 10 (mod 19).…

A: (d) Given is a congruence 9x+13≡10(mod 19)   To Find: An integer x such that 0≤x≤18 that satisfies…

Q: (a) Suppose that rị and r2 are quadratic non-residues modulo an odd prime p. Does x = r¡r2 (mod p)…

A: Quadratic Residues: Let m,p=1. m is said to be the quadratic residue mod p if the following equation…

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: Find all incongruent solutions - if exist - for the congruence: 36x = 96 mod 156

A:

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

A:

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: Can you please explain to me why is (k(k+1) ) / 2 is an integer and I know it always even so can you…

A:

Q: Give the smallest extension of Q that contains all the zeros of (x4 – 8x² + 15)(x² + 1). -

A: Yes

Q: Solve by using the Chinese remainder theorem for the congruence X 2 (mod 7) X 4 (mod 5)

A: Given that This can be written as,

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

A:

Q: 2. Find all solutions, or show there are none, to the congruence X78 = 5 (mod 19)

A: By Fermat little theorem x18≡1mod19. Therefore, x4×18+6≡x6mod19 or x78≡x6mod19.

Q: Prove that (3+2/5) is irrational

A:

Q: Solve the congruence x² = 9 (mod 600).

A: Given congruence is x2≡9 mod 600

Q: Find all solutions of the following linear congruences, Inverses (when required) may not be…

A: Given: 14x≡6(mod 31) Here a=14; b=6 and N=31 Also gcd(14,31)=1 Hence there can be only one value of…

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: Suppose that a, b. are two integers such that 3a² – b = 1. Does it follow that a and b are…

A: Given that,

Q: classify the following PDES :- - 227 827 -2 +2 82 дуг =x+3y дх2 ахду 2 2 от Ихх - 2ху Иху +у Иуу -…

A:

Q: 7. Suppose a1, a2,...,an are integers, r is a solution to the equation x" + a1x"- + a2x"- +.. + an =…

A:

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: Obtain the expansion of : 2 up to the term in x8 1-3x2

A:

Q: Use the construction in the Chinese Remainder Theorem to solve the system of congruences x = 2 (mod…

A:

Q: ulus) 1) Solve the following congruences system. 5x +4 = 12 (Mod 7) 14x - 3 = 19 (Mod 5) 2) Prove…

A:

Q: Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a non-negative integer

A: We need to prove by mathematical induction that, 3 divides 17k+1-2*5k+3+11k+3 whenever k is a…

Q: Prove that ifx^2−6x+ 3is odd, then x is even

A:

Q: 10. Solve the following modular congruences or show that there is no solution. (^) 10r =1 (mod 31)…

A: We will find out the required value.

Q: The number of mutually incongruent solution(s) of 307x = 83 (mod 360)

A: BY THE DEFINITION OF Modulus congruence Modulus congruence means that both numbers, 16 and 21 for…

Q: (b) Since 61 is a prime with 61 = 1 (mod 12) the polynomial X2 +3 has roots in Z/61Z by (a): they…

A: Given that, 61 ≡ 1 mod (12) The polynomial x2 + 3 has roots in Z/61Z. The roots are 27 + 61Z and -27…

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

A:

Q: Ф(тk) — ф(0) — тk

A:

Q: 4. Verify thatf (x) = x³ – x2 – 20x + 6 satisfies the three conditions of Rolle's Theorem on [0,5].…

A:

Q: Use the Chinese Remainder Theorem to find a solution to the congruences х 2 тоd 10 x 3 mod 1001 х 3…

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: e the Chinese remainder theorem to solve the system x = 3 mod 5 x = 5 mod 11

A:

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

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.