Find all integers x + 3 such that x – 3 is divisible by x – 3. |

Q: Prove that (-1+ iE)" + (-1- i5" has either the velue 2n+! or the value -2" f ロ ww. š any integer…

A: The cube root of unity is written as:

Q: Find all integer x satisfying the condition x = gcd(4x + 3, 2x). Justify your answer.

A: To find all integer x satisfying the condition  x=gcd4x+3,2x

Q: n -r Prove that („",) : r+1

A: Combinations: Combination is the selection of n items taken r at a time is

Q: Prove that for all positive integers k and n, with k < n, n-1) k-1/ k 1 k ... k k- 1 k- 1

A:

Q: 3. Prove that for all x > 0 and all positive integers n, x3 et >1+x + 2! 3! n!

A:

Q: Find all positive integers n such that ϕ(n) = 12. Be sure to prove that you have found all…

A:

Q: n Prove that + 2 + 22 +...+ 2" +...+ 2n = 3" for all integers n 2 0. * ^) xn-ryr and x=4=1.…

A: We have to prove given identity:

Q: Prove that (1 + x)n≥ (1 + nx), for all natural number n, where x &gt; – 1.

A:

Q: 2. Let an = 3n + 2 for n = 0, 1, 2, ... (i ) Find ao, a1 and a2. (ii ) Show that a2 = 4a1 – 3ao.…

A: Answer and explanation is given below..

Q: Letf(x)= (x - 1)(x - 2)(x - 3) Prove that there is more than one 'c' in (1, 3) such that f' (c) =0.…

A: We have to find the value of c

Q: 8"-3 is a multiple of 5. In other words, prove that 8"-3"=5 for some i integer j.

A:

Q: Prove 1 +,+ i++ = 2 – for all positive integers n. |

A: We want to prove 1 + 12 + 14 + ... + 12n = 2 - 12n

Q: Prove, for all positive integers n, 1 3 2n – 1 1 - 2 4 2n (3n

A:

Q: Prove by mathematical induction that 3n+2n+3 is divisible by 4 for all positive integers

A:

Q: Find all integers n > 1 such that 1n − 1)" is +2+...+(n divisible by n.

A: To find all the integers n&gt;1 such that 1n+2n+...+n-1n is divisible by n

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

A:

Q: find all real such that 12x-11>x

A: The inequality can be calculated using absolute values. The absolute value function takes a positive…

Q: 1. The set of all pairs of real numbers of the form (x, 0) with the standard operations on R2.

A: 4. Given that the set of all pairs of real numbers of the form (x, 0) with the standard operation on…

Q: Prove that the following: 2"*(-1)"+1 (- 1)"(2)" = for all real non-negative integers (n) 3

A: Given :- The discrete problem -1n 2n = 2n+1 -1n + 13 for all real non-negative integers n.

Q: Use the Factor Theorem to prove that x + c is a factor ofxn + cn if n Ú 1 is an odd integer.

A: To prove the given statement.

Q: Find all integers n such that n| (2n+3). *

A: Given, n|(2n+3)     TO find, All integers n

Q: Let a = 1, and let an+1 4a, – 1, for all nonnegative integers n. Prove that an 2-1"+1 3 %3D

A:

Q: Find the supremum and infimum of the set A := ∩ ∞ n=1 (1 −1/n ,1 +1/n ). And prove it

A:

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 &amp;6 For detail...go through the solution

Q: HW4| Prove How 4 Prove if x;→ × then 6x.→6x Xj ->

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

Q: Prove that x2 + 3x + 3 ≥ 0 for all x.

A: We have been given the quadratic equation  x2+3x+3

Q: Use the factor theorem to verify the statement

A: Consider the statement  x−y is a factor of xn−yn for every positive integer n.

Q: Determine all integers n ≥ 3 such that Cn is planar.

A: ANSWER :

Q: (a) Prove that for all integer n 2 1, 3|n(n² + 11).

A: a.          In the given problem we have to prove that for all integer n≥1, 3|n(n2+11). To prove…

Q: A2

A: Consider the matrix A is, A=0100 Apply the condition, A2 = O. 0100·0100=0000 The matrix A is…

Q: Let p be a prime. Show that xpn −x+1 ∈ Fp[x] is irreducible iff either (1) p = 2 and n =1,2 or (2) p…

A:

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

A: (a) Consider two odd integers 2m+1, 2n+1 such that m+n is even. Now, Consider the average of the…

Q: n Prove that (3i – 2) = ; (3n – 1) for all integers n > 1. - i=1

A:

Q: 2""(-1)" +1 2. Prove that 1-2+2² – 2³ +...+(-1)" 2" =- for all positive integers n.

A: We prove this result by using mathematical induction. For answer please look at step 2.

Q: Prove that (f^2)_1 + (f^2)_2 + ⋯ + (f^2)_n = f_n*f_(n+1) when n is a positive integer.

A: We have to show that  f12+f22+f32+.....fn2=fnfn+1 where n is a positive integer and fn denotes the…

Q: 8. All polynomials in P, such that p(0) = 0.

A:

Q: let a>0. prove that if x is a number such that |x-a| a/2

A: Given a&gt;0 and x is a number such that x-a&lt;a2 We will use the property If y&lt;b , then…

Q: Find the smallest integer p so that f(x) = −2x3+8x+2 must have a zero on [0, p] 2,1,5,4, or 3

A: Note: We are not given any particular method. So, we will be solving by the method of graph for…

Q: Find all positive integers n such that p(n)|n.

A:

Q: Letf(x) = (x - 1)(x - 2)(x - 3) Prove that there is more than one 'c' in (1, 3) such that f' (c) =…

A: We have to find the value of c

Q: 7. Show that each polynomial f(x) such that f(x) is irreducible in Z, (a) 7x + 6x? + 4x + 6 (b)

A:

Q: 2n° 1. Prove that for all positive integers n > 1, the inequality 2" < <(") holds. п

A: Consider the given statement be P(n) as; Apply induction methods to prove the above statement P(n)…

Q: 2 Lk=0 4. Prove that for all positive integers n, п

A:

Q: Determine the number a in Z5 so that (x - a) becomes a divisor of f (x) = x ^ 3 + 2x ^ 2 + 3x + 3 in…

A:

Q: a) Prove, for all integers k, that 2k-1 is odd. b)prove, for all integers x, if x is odd, then x =…

A:

Q: Use the factor theorem to verify the statement

A: Given statement is, x+y is a factor of xn-yn for every positive even integer n.   So, Let fx=xn-yn,…

Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.