There exists two odd integers whose average is odd. 1 1 1 There exists distinct positive integers a, b, c such that a

Q: Prove that for all positive integers n, we have 6″ – (-4)″ Σ (1) 5k 2 k−1 k odd =

A: Expand 1+xn 1+xn=C0n(1)nx0+C1n(1)n-1x1+C2n(1)n-2x2+.......+Cnn(1)0xn=∑r=0nCrnxn Expand 1-xn…

Q: 14. Let (a, b) = 1 and 2 = y for some integers x and y. Then prove that there exists an integer n…

A:

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

A:

Q: Prove that for all positive integers, 1 1 + 3.4 1 ... 1.2 2-3 n(n + 1) n+1'

A:

Q: Let n be any integer. Prove that 3n - 2 is odd if and only if n is odd. Suppose x and y are any…

A:

Q: %3D If there exist integers r and s such that ra + sb = d. then d = (a, b). lgn hi O

A: Given statement is false

Q: Find an integer n such that U(n) is isomorphic to Z2 ⊕ Z4 ⊕ Z9

A:

Q: Given positive integers k and m, there exists an integer N ≥ m such that N →(m)k

A:

Q: a. Prove that the v63 iş irrational b. For all integers, prove that 4 (n2 + 2) c. Prove or disprove:…

A: Note: As per our company guidelines we are supposed to answer the first three sub-parts of the…

Q: 2n+1+(-1)" Prove that for all integers n 2 0, 3.2"

A:

Q: Find a positive integer m > 3 such that there are at least 3 integers n in Zm such that n2 = 1.

A: We have to find a positive integer m≥3 such that there are at least 3 integer n in Zm such that n2=1…

Q: 4.71. Let n E Z. Prove that 5 n if and only if 5 n. 4.72. Prove that if a, b and c are nonzero…

A:

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: Prove there exist no integers a and b for which 18a + 6b = 1

A:

Q: Prove the claim that there are no integers j and k such that j^2 + 3k^2 = 8

A: Since both j and k are being squared so if we prove the case for positive integers, that will prove…

Q: true or false In the domain of Gaussian integers Z[i], the element 17 is Irreducible

A:

Q: Prove that there exists two integers a and b such that a+b > ab

A: We want to prove that there exists two integers a and b such that a+b>ab

Q: 26. Let E be the set of all even integers, and let O be the set of all odd integers. Let X = {n €Z|n…

A: If y is an even integer then it will be of type 2n for some n in integer If y is an odd integer then…

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: 8. The identity element of negative integers (Z-) under the operation * defined as a*b = min(a,b) is…

A:

Q: 37. There exists no positive integer x such that 3x < x <4x. If x is irrational and y is rational,…

A: 37. Given statement: There exists no positive integer x such that 3x&lt;x2&lt;4x. If…

Q: Prove: If m and n are non-zero integers and m > n, then there exists an integer q such that (1/n -…

A: Let m and n are non-zero integers such that m&gt;n To prove there exists an integer q such that…

Q: 3. Let n e Z be nonzero and even. Prove that there exist unique a € {0, 1}, b € N, and odd c€ N such…

A: Since n is even and non-zero . Since n is non zero ⇒ n is either positive or negative. Let's write…

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: Let k and n be two positive integers, and assume that n is odd. Prove that there exists an integer a…

A: Given: Let k and n be two positive integers in that, we assume if n is odd.

Q: Let a and þ be integers, not both zero. Then a and b are relatively primeif and only if = ax + by,…

A: I have given counter example for each false option

Q: Let x, y, and z be positive integers. Suppose that x2 + y2 = z2. Prove the following. (a) if z is…

A:

Q: 14. Let (a, b) = 1 and x" = yb for some integers a and y. Then prove that there exists an integer n…

A:

Q: 1. Prove the following using direct proof. a. If 5x – 3 = x + 5, then x = 2. b. If x and y are both…

A:

Q: Prove that gn = 5 · 3n + 7 · 2n for all integers n ≥ 0.

A:

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: 3. Let n e Z be nonzero and even. Prove that there exist unique a € {0, 1}, b e N, and odd c e N…

A: Consider the given condition,  n∈ℤis a non-zero and even. The objective is to prove that there exist…

Q: Let A = {4n^2 + 1 | n ∈ Z}. Show that the size of A is equal to the size of Z≥0. Z≥0 is the set of…

A: A set A is said to be countable  if there is a bijective map between A and set of natural numbers .…

Q: Prove by any means that n!/(n − r)!r! is always an integer for 0 ≤ r ≤ n

A: Since nCr = n!/(n-r)!r! So given expression represents combination and it is read by "n chose r"…

Q: (6) Suppose A CR is countable. Show that there exists a real number r such that An(r+ A) = 0. (Here,…

A: We prove this by contradiction

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

A:

Q: Given positive integers k and m, there exists an integer N ≥ m such that

A:

Q: 1| = 1 (12 = 3 ak = ak-1 + ak-2 for k23 () Prove that an 1.

A:

Q: Let a and b be integers, not both zero. Then a and b are relatively prime if and only if 1 аx + by,…

A: Option (2) is correct.

Q: 18. a. Prove that for every integer a, if a' is even then a is even. b. Prove that V2 is irrational.

A:

Q: Let n be a positive integer. Prove that for all x € Z, if x" is odd, then x is odd.

A:

Q: Prove that 1 n i(i+ 1) n+1 i=1 whenever n is a positive integer.

A:

Q: .3. a. Prove that for every integer a, if a' is even then a is even.

A:

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

A: Given a0 =1 an+1 = 3an +2 for all non negative integers n. We need to prove an = 2.3n -1

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

A:

Q: 1. Suppose a e Z is an integer. Prove that a is odd if and only if a³ is odd.

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: Prove that for all positive integers n, we have 6″ Σ (1) * _ "(-¹)" 2 k=1 k odd IG

A:

Q: 2. Find a positive integer a such that 4 | a, 9| a+ 1, and 25 | a + 2.

A: