4. Given a, b € Z, such that mdc(a", b") = 1, \n € N₁ n ≥ 2, Verify that a and b are prime numbers between each other.

Q: n 2n + 2 /2 n 2n k k=1 \k –1, п+1 n -

A: The objective is to prove that ∑k=1nnknk-1=2n+2n+1/2-2nn This theorem can be proved by induction,…

Q: 2. (a) Find integers a, m, n such that m | a and n | a, but mn /a.

A: .

Q: 6. Let a be a number written (in base 10) as ao 10⁰+ a₁ 10¹ + a₂. 10²+...+an. 10" where 0 ≤a < 10.…

A:

Q: For positive integers m > n , show that -2 + n- 1 -3 + ...+ n + n-1 n - 1 + in

A: The right-side of the given equation can be expressed as…

Q: |Let A¡ = [i − 1, i] for all integers i = 1,2,3,4. Find the following. a. A₁. A₂- b. A₁ U A₂ U A3 U…

A: The given set Ai=i-1, i for all integers i=1, 2, 3, 4. We have to find, a) A1, A2, A3, A4 b)…

Q: 3.) Is the following Credit Card number valid? 4716 5868 1970 2056 1) Find the check digit (d) for…

A:

Q: 5. (*) If a and b are nonzero integers such that a | b and b|a, prove that a = ±b.

A: .

Q: 20. Which statement represents the relationship between P = {9, 11, 13, 15, 17) and Q = (2, 4, 6, 8,…

A: As per guidelines we are supposed to answer first question only please repost the Other question.

Q: Let S be the set of all real numbers except -1. Define * on S by a*b = a + b + a b. The solution of…

A:

Q: 5. Show that 2 32 3" 31 3" for all positive integers n.

A:

Q: Prove that n! > 2" for n 2 4.

A:

Q: Prove that 6 divides n³ - n whenever n is a nonnegative integer.

A: In this problem we have to prove that  divides  where  is a nonnegative integer.

Q: n+1 sup { , ne N} = 1. Select one: True False

A: The given set is                                  n+1n, n∈ℕ

Q: Find a number n e N such that 2" > 102000 and 2" has among its 2000 last digits at least 1000…

A: We will use logarithm to solve the problem

Q: 5. Let a be a real number, c an integer, c > 0. Prove that

A:

Q: Show that for integers a, n satisfying 0 < a <n, ged(a, n) = gcd(n, n-a)

A: Solution is given below:

Q: 33. Let R be the set of all real numbers, x ER and n eN a n 2 2. If (1+ x) > 0, then which of the…

A: Option (C) II only is correct.

Q: Let h be a prime number and let k be some integer. What are the possible values of ged(h+k, h-k)?

A: We have given that ,  h be any prime number and k be some integer. We need to find possible values…

Q: Let c, a, B be nonzero real numbers such that c + B. Find a given that Σ 142п -з 15 ch–4 ac? с — В…

A: The sum, S of the infinite series is calculated using the formula, S=a1-r , where a is the first…

Q: Evaluate |1 – cos40 de O 1/(2v2) V2/2 O None of the choices O 1/2

A:

Q: 3. Let a₁ = 2, a2 = 1, and for each natural number n > 2, let an 2an-1-an-2. Prove that for all…

A:

Q: 16. For any two integers a and b, (a + b)² = a² + b² FALSE; Gir

A: First, let's understand the equation. The equation given is (a + b)² = a² + b². This equation is a…

Q: Let n 2 2. Assume that Fn is prime. show that -1.

A:

Q: Let i be a positive integer and A, = [-i+1,1]. Then U2, A

A: We have to use properties of union in this question.

Q: 21. How many positive integers x are there such that 3x has 3 digits and 4x has four digits?

A:

Q: 10. Let a and b be natural numbers that are co-prime. Prove that (b-a) and b must also be co-prime.

A:

Q: Let A = {n E Z n = 5r for some integer r} and = {m e Z | m = 20s for some integer s}. Prove or…

A: We need ton prove or disprove given statement. We will prove in general and To Disprove we need only…

Q: : a number is even  if

A:

Q: Show that (n + 1)! > 2"+3, for n > 5.

A:

Q: Let, a1= 3 and for n ≥ 2, an = 2an-1 + 1, express an in terms of n.

A:

Q: 9. Let A = 0100 001 0 0001 0000 to Show that A" = 0 for n ≥ 4. 6]

A: Solution:

Q: (14) Let E denote the set of even integers, O denote the set of odd integers, and x, y € Z be any…

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

Q: 12. Let the "Fibonacci-2" numbers g, be defined as follows: 91 = 2, 92 = 2, and gn+2 =( In+1)( In)…

A:

Q: (b) Prove that E-o(-1)* )= 0 for all positive integers m.

A: We have to prove ∑k=0n-1kkn=0 for all positive integer n

Q: 11. Establish the following relation between Fibonacci and Lucas numbers: L, = un + 2un-1 for all n…

A: Remark: We know that both the Fibonacci numbers and  Lucas numbers are obtained as the sum of its…