(c) Prove that m is prime if and only if y(m) = m - 1.

Q: Show tht if x is an integer, then gcd(x+1, x2-x+1) divides 3.

A: Given that x is an integer. To show :(a) gcd (x+1,x2-x+1) divides 3 (b) To determine for which…

Q: Show that x4 + 1 is reducible over Zp for every prime p.

A: In this question, we have to show that x4+1 is reducible over ℤp for every prime p.

Q: Determine (with proof) if there exist integers m and x such that 2003m 2 = 2.x(x – 20). If they…

A:

Q: Suppose that there exist integers x and y such that ax + by = 1. Show that a and b are relatively…

A:

Q: Prove that 10n +9 and 6n + 5 are relatively prime for all n E Z.

A: Two numbers are said to be prime  to each other if they  share the gcd 1. That is to say  suppose…

Q: Let p be a prime.a. Determine the number of irreducible polynomials over Zp of theform x2 + ax +…

A: For any fixed prime p, we have to determine the following : (a) The number of irreducible…

Q: Let p be a prime that does not divide n. Prove that Φpn(x) =Φn(x p)/Φn(x).

A: Given: p be a prime number that does not divide n. To Prove: Φpnx=Φnxp/Φnx.

Q: Let a and b be positive integers and let b= qa +r, where 0 ≤r <a. Prove that (a) 2b - 1 = 2" - 1…

A: Let  and  be two positive integers.Let  where (a) We  have to prove that      Now, observe that…

Q: Using Fermat's little theorem, show that if p is prime and (a) if x-1 + yp-1 = zp-1, then p | xyz.…

A:

Q: a. Is (x - 1) a factor of x" c. Is (x - 1) a factor of x" - 1? e. If n is a positive even integer,…

A: Hey, since there are multiple sub-parts posted, we will answer first three sub-parts. If you want…

Q: Let p be a prime, a ∈ Z. Prove that ap − a is divisible by p.

A:

Q: "EZ¹, 2n + 3 is prime."

A: We have to prove or disprove the given statement using given information.Concept required:1) One…

Q: 4- Let n be a positive integer, and define T 2 -1. Prove that T, has at least n different divisors.…

A:

Q: If z1 and zą both satisfy z +z = 2 |z - 1| and arg (21– 2): then find Im (21 + z2). 4'

A: let z1 = x1+iy , z2 = x2+iy2

Q: a) What is the generating function for {ak}, where akis the number of solutions of x1 + x2 + x3 = k…

A:

Q: Consider the following proposed proof, which claims to show that every nonnegative integer power of…

A: Note: we know that in strong induction principle, we assume that the particular statment holds at…

Q: Let n and k be integers such that (n, k) = 1. Prove that (n, n − k) = 1.

A:

Q: By using Caushy Rieman prove that (2) and (3) (2) 4c2) = (3) prove That (3)- 3.

A:

Q: Show that for the Legendre polynomials Pn the following holds for all n E N (n+1) Pn+1(x) − (2n…

A:

Q: Find {2), f3), R4), and {5) if fis defined recursively by f0) = -1, A1) = 2, and {(+1) = Jtn}*s(n –…

A:

Q: 5. Prove that p(mk) = mk-1p(m) for all positive integers m and k.

A:

Q: Show that n is composite if and only if p(n) <n – Vn.

A:

Q: Prove that 3 and 1+v-5 are relatively prime in Q(V-5), but that there exist no integers A, µ E…

A: We will use the basic knowledge of ring theory and UFDs to answer this question properly, in detail.

Q: Prove that: n n (3) + (-²)=(-+ m m+1 n+1 m+1

A: We have to prove the given identity

Q: (5) {1,2} C P({1, 2}).

A: Given  set is {1,2},  we knew that power set of a given set is set of all subsets of the given set.…

Q: f(z)=z²-(6+-4i) z+(-27+12) has 2 roots: z₁ and z2, sorted in an increasing manner according to the…

A:

Q: Suppose that m, n are two integers such that m3 – 2n2 = 1. Does it follow that m and n are…

A:

Q: Show that t (T (n)) = 2 if and only if n = p9-1 for some primes p, q. %3D

A: T(n) is the divisor function which counts the number of divisors of a number.

Q: Use Fermat's theorem to prove that if (n, 23) = 1, then 23l 110-1. Use Fermat's theorem to prove…

A: The given problem is to prove the following identities using fermat's theorem and it's properties.

Q: ak + b) {{} = m – 1). k=0

A: Please cheek the details in the next step

Q: Give an outline for a proof of "for any integer r and any positive integer d, if d|(x– 1), then d…

A:

Q: n (3n–1) 2 B. Use mathematical induction to prove that P(n): 1+4+7+..+(3n-2) = n2 1. Use the first 2…

A:

Q: (b) Show that if gcd(m, n) = 1, then σ₁ (mn) = σt(m)ot(n). In other words, show that function. Is…

A:

Q: Determine whether the polynomial 4?10  − 9?3 + 24? − 18  ∈ ℤ[?] satisfies an Eisenstein criterion…

A: Let ? = x, then, 4x10  − 9x3 + 24x − 18  ∈ ℤ[x]

Q: For any integer k greater than or equal to 2, define R(k) as the ratio of k 's largest positive…

A: Let's break down the problem into two parts: (a) For every natural number 𝑥, there is a natural…

Q: Let U = {x: x is an integer and 2 7}, B = {x: x² > 20}. ii. A = {x: x2 - 3x + 2 = 0}, B = {x:x + 7…

A: We have to determine which condition applies on set A and B.

Q: Use the factorization A = PDP' to compute A", where k represents an arbitrary positive integer. -2…

A: