Q: If possible, let E be rational.Then, a there exist positive co - primes a andb such that 12 b

Q: m and n be positive integers so that m n. Show for any integer a that the congruence

A:

Q: (b) Prove that (a, b) = (b,r).

A:

Q: Prove that for any two rational numbers r and s, the sum r+s is rational. Paragraph Lato (Recom... v…

A: I have given the answer in the next step. Hope you understand that

Q: Give a formula for the nth positive even integer: Here is a theorem: 1+ 2+3 + ... + n п(n+1) n = Use…

A:

Q: Prove the following using the specified technique: (a) Let x and y be two real numbers such that r…

A: For part a: The objective here is to prove that if x is irrational, then x-y is irrational. Given: x…

Q: Suppose x is a positive rational number and x? < 2. Let y = 22. Prove that x < y and y? < 2. X+2

A: We will use basic knowledge of set theory and logic to answer this question.

Q: 3. x* + 1 is reducible over Z, for every prime p. a) True b) False

A: We have to find out whether x4+1 is reducible over Zp for every prime p or not a polynomial is said…

Q: Prove the following using the specified technique: (a) Let z and y be two real numbers such that a+…

A:

Q: 2. Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? +1 is divisible by 10. 3.…

A:

Q: 3. Show that for any rational number r and any X>0, , In x= rln x'

A: The second fundamental theorem of calculus  (Newton Leibnitz formula) states that  : If h(x) and…

Q: Let a and b be two integers such that b | a. Let p be a prime such that p|a and p{b. Prove that p|…

A:

Q: d) If n iš a If t denotes the nth triangular number, prove that in terms of the binomial…

A: The binomial coefficient, nr is defined using the formula, nr=n!(n-r)! r! . The binomial expression,…

Q: Let a,b,c, p be rational numbers such that p is not a perfect cube. If a + b p3 +cp = 0, then prove…

A:

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

A:

Q: If p and q are odd primes, then a)-4 is a primitive root of q. b) 4 is a primitive root of q. c)…

A: Given p and q are odd primes.

Q: For any distinct primes p, q,r and s, we have pq # rs. For any positive integer n, there exists n…

A:

Q: 6. Use the argument in the proof of Theorem 2.1.4 to show that there does not exist a rational…

A:

Q: (1) Prove that is a, b, and c are integers with c + 0 then ac = bc = a = b ( that is this a key step…

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

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: 6. (i). Prove that every prime > 3 is of the form 6m +1 or 6m + 5.

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: Let x be a real number. Prove the following statement: for each e > 0, there exists a rational…

A: Given  that x be a real number. And we have to prove the following statement: for each ε&gt;0, there…

Q: Given: "if x is rational and x + 0, then 1/x is rational", can be proven by Direct proof. O True…

A:

Q: a) Verify this equation using mathematical induction that for all n2 1, n(3n – 1) 1 +4+7+ ·.. + (3n…

A:

Q: If n > 2 is prime then n | a"-1 1 for all positive integers a with gcd(a,n) = 1. Select one: O True…

A: True

Q: This are a Math discrete structure questions.

A: Negation of If A, then B is A and not B A or B is not A and not B A ^ B is ⌐A v ⌐B

Q: (a+bVp|a, bEQ; Pa prime). 0. a) Consider the subset SCR' defined by S - Show that (S,+, ·) is a…

A:

Q: Let n be an integer. Apply any methods to prove that n? can be written as either 3k or 3k + 1 for…

A: By the quotient remainder theorem, any integer n can be written as n= 3q+r, that is n=3q or n= 3q+1…

Q: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p is not prime in Z[i]

A: Given that p is odd prime. This implies that p is not equal to 2.

Q: Prove the following using the specified technique: (a) Let r and y be two real numbers such that r+y…

A:

Q: Let n E Z', where n > I. Show that if n is a product of distinct Mersenne primes, then o(n) = 2k for…

A:

Q: (a) Characterize the primes p for which 63 is a quadratic residue.

A: Quadratic residue: Let p be any odd prime member and gcd(a, p)=1. Then a is said to be a quadratic…

Q: Let n 2 1, a be a real number, and r -1. Prove the following statement using mathe- matical…

A:

Q: If S is a nonempty subset of the negative rational numbers, then: 3x e S(Vy e S(x > y)). O true O…

A:

Q: Let n 2 1, x be a real number, and r > -1. Prove the following statement using mathe- matical…

A:

Q: 19. If x is a real number such that x' <0, then you are the Pope. If ris a real number such that ris…

A: True. For any real number x, square of it is always a positive number (x2&gt;0) For any x, if…

Q: Let 5 be a primitive 9th root of unity in C and consider the extension Q(5) of Q. Using the formula…

A:

Q: If S is a nonempty subset of the negative rational numbers, then: Jx e S(Vy E S(x > y)). true O…

A: Note: Rational numbers are countably infinite. So there is no largest rational number and no…

Q: If |G = 30 and |Z(G)| that |G| 5, what is the structure of G/Z(G)? What is the structure of G/Z(G)…

A: Solution:-

Q: Prove the following using a proof by contrapositive: Let z be a rational number. Prove that if zy is…

A: A number x  is said to ration if there exist two integers  p and q  such that  x=pq

Q: Show that the three properties listed  are valid for Zp,where p is prime. a. a2 = a implies a = 0 or…

A:

Q: For every prime p, show that Х-1 — 1 %3 (х — 1)(х — 2) : .. [x — (р — 1)] in Zp[x].

A: Fermat's Little Theorem: Let p be a prime and let a∈N , 0&lt;a&lt;p. Let Ap={1a,2a,....(p-1)a} No…

Q: latively prime if, and only if, either a or b is prime. there are integers r and y such that ar + by…

A: Two positive integers a and b are relatively prime if and if only

Q: – $ (Ker ()) is: a) Prime ideal b) Maximal ideal c) Both d) None

A:

Q: Use the argument in the proof of Theorem 2.1.4 to show that there does not exist a rational number s…

A:

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: (a) Let p be a prime, and let a be an integer. Show that a is relatively prime to p if and only if…

A:

Q: 7. Suppose a1, a2,...,a, are integers, r is a solution to the equation x" + a1x"-1 + a2x"-2 +... +…

A: for question number 7  we use contradiction to prove the statement.

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: let us assume that /3 is a rational number .show that V3 = where a,b are co – primes b

A: