[Proof] Prove that the congruence (x² − 2)(x² – 19) (x² - 38) = 0 (mod p) has a solution for every prime p.

Q: (2) Use induction to prove the following for all natural numbers n: 1+2+ 22 + 2³ + + 2" = 2"+1 – 1.…

A:

Q: a) Give an example of a number m possessing exactly 12 primitive roots. b) How many primitive roots…

A: As per company guidelines I have solved first three sub parts if you want to get answer of the 4th…

Q: Show that for any integer n, n² – 2 is not divisible by 3.

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

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: i +1 If z = E C, find simple expressions for i and /2 in terms of z. Then prove: If p is an odd…

A: Given, Z=i+12∈ℂ We can write the above equation as, Z2=i+122=i2+2i+12=-1+1+2i2=iSo, i=Z2

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: (a) Find three distinct integers satisfying the equation r = 1 mod 11. (b) Prove that if k is an…

A: a≡bmod m means a-b is divisible by m where a and b are integers. For example 13≡3 mod 5⇒13 - 3 is…

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: 1. Solve the congruence x² + 1 = 0 mod 13³.

A: According to the given data, find the solution:

Q: Proficiency #9. [Modular Arithmetic] (a) Find integers x and y such that r 0, and x² + x +1 = y? +…

A: We will solve the following two questions in the next step.

Q: (9) Find all prime numbers p for which both equations X²-2 0 mod p and have solutions. X²-3 0 mod p

A: Concept: The quadratic congruence: x2≡a(mod p) , where gcd(a,p)=1 Has a solution if (ap)=1 That is a…

Q: [Formal Proof] Let x in Z. Prove that 7x + 3 is even if and only if 3x - 4 is odd.

A:

Q: 1. Find all simultaneous solutions of the congruences: (a) X= 3 mod 5 x= 4 mod 6

A:

Q: Let f(x) be a fixed integer polynomial, and let m be a fixed positive integer. Denote the number of…

A:

Q: a) Find all solutions to the congruence 10r = 5 mod 15

A: Given,

Q: Icm(a,b) = ab if and only if a and b are relatively prime.

A: We know the relation between product of the two numbers their L.C.M and G.C.D Let a and b two…

Q: Let Z be the set of all integers and Q be the set of all rational numbers and R be the set of all…

A: As per our guidelines I solve first 3 subpart, so kindly request you to repost the remaining…

Q: Prove that for all integers m and n, if m is odd and the remainder is 2 when n is divided by 3, then…

A: Given, m is odd  When n is divided by 3, then the remainder is 2  We have to prove when  3m+2n  is…

Q: (a) There exists two odd integers whose average is odd. 1 1 (b) There exists distinct positive…

A: (a) ans:- 1,5 (b) ans:- 2,3 &6 For detail...go through the solution

Q: ove or disprove: There exists an integer a such that 14 (3a-5) and 21 (2a +

A:

Q: Let p be an odd prime. Show that if (p − 1) Xn, then 1 + 2 + + (p − 1) = 0 mod p

A: The given problem is to show that for p an odd prime if (p-1) not divides n then 1n+2n+...+(p-1)n=0…

Q: Let f(x) be a fixed integer polynomial, and let m be a fixed positive integer. Denote the number of…

A: We use basics of set theory to solve this problem. The details are given as step 2.

Q: the congruence + 92 (2), using the method (Bertrand's Postulate). If n > 0 there erists a prime p…

A: Given:- x4-2x3+9x2+4≡0 (mod 72) , using the method (Bertrand's Postulate).  if n>0 there exists a…

Q: Let a and b be odd positive integers. If there exist integers s and t with as+bt=4 then gcd(a,b)=4 a…

A: We solve given problem with the help of Extended Euclidean Algorithm

Q: Prove: An odd plus an odd equals an even. (2m + 1) + (2n + 1) = [ ? ]m +[ ]n+ [ ] = 2(m + n + [ ) =…

A:

Q: (b) Prove that if a = 7 mod 10 and b = 6 mod 15, then a + b = 3 mod 5.

A: Our objective is to prove . Since you have posted multiple questions I am solving one.

Q: (a) Use proof by contradiction to show that there is no positive real number r such that x-2 X > X…

A:

Q: Prove that if p and q are distinct odd primes, then we have (p-1)(q-1) 4 25 52 TID = (-1) quadratic…

A:

Q: .. (a) Prove by mathematical induction that 9**2 – 8n – 17 is divisible by 64 for all positive…

A:

Q: Lef 5={x|(x-)(x-3)< of; T= }x| 12x+1)s a) Show s & T on Ore musndn line. 6) Are S& T or cloed on…

A:

Q: (12) What values can be assigned to r so that {4n+r} will include all odd primes?

A:

Q: Factorize (i) 1144 in Z (i) 1144 in Z[i] (ii) 1144 in Zp[x] where p is a Z-prime greater than 1144.

A: Given

Q: (24) Let p, q € N be two distinct primes. (a) Find the following quantities (possibly in terms of p…

A: Given: p, q∈ℕ are two distinct primes. To Do: (a) Find gcdp, q,   gcdp2q, pq2,   gcdpq, p2q2+1. (b)…

Q: The sequence fn is given asf1=1 ,f2=3 ,fn+2=fn+fn+1 for n>_ 1. How many even terms are between the…

A: Calculate first few terms of the sequence, by substituting, n=1,2,3,... in the equation,…

Q: Given a permutation S={4,5,7}, Find: a.) its inverse b.) even and odd permutations

A: