For any non negative integer n , prove that an – bn is divisible by a-b,where a and b are any integers with a ± b.

Q: Prove that if a and b are integers with a 0 and where + b - a = 0, x is a positive integer such that…

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Q: Abstract Algebra. Please explain everything in detail.

A: To prove (or disprove)  the possibilty of required representations under the given conditions

Q: Show that every positive integer is of the form 2q,and that every positive odd integer is of the…

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Q: Prove that if a is an odd integer, then 5a² – 3a + 5 is odd.

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Q: Prove by direct proof that any integer n ≥ 3 can be part of a Déjà vu Pair n = [11 · · · 1]b for…

A: Given: Any integer n≥3 can be part of a Déjà vu Pair n=11...1b for some base b≥2.   To prove: The…

Q: Prove that for any integer a, 3 divides

A: To prove that for any integer a, 3 divides (a-3)(a+4)(a+2).

Q: For any integer k, prove that gcd(2k +3, 5k +7) = 1.

A: Use the given information.

Q: Prove that for a positive integer n we have that  33n+3 - 26n - 27 is divisible by 169.

A: To show  33n+3 - 26n - 27 is divisible by 169.

Q: Show that 3 · 4n + 51 is divisible by 3 and 9 for all positive integers n.

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Q: let b be a nonzero integer. prove that gcd (0,b) = |b|.

A: We need to prove that for a nonzero integer b,

Q: (e) For any nonzero real number z, prove Σ''-1 1-z"

A: The objective is to prove that for any nonzero real number x, ∑i=1nx-1xi=1-x-n

Q: Let a, b be two positive integers with (a, b) = 1. If ab = c, then prove that there exists

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Q: For any integer a, prove that a = 0 or 1 (mod 3).

A: Every number is either of the forms 3k, 3k−1, 3k+1 where k is an integer

Q: show that for eury A s integer nz3; A n- Am-2 +A2J. Mence determine ASO.

A: To show: That An=An-2+A2-I for n≥3 then find A50. Given: The matrix is A=100101010. Calculation:…

Q: Suppose x and y are any integers. Prove that if x and y are odd, then x + 5y is even

A: Suppose x and y are odd integers. To show x+5y2 is even. Note: Square of an odd number is odd.…

Q: Let a and b be positive even integers. Prove that (a, by 2

A: Here, a,b denotes the gcd of a and b. a2,b2 denotes the gcd of a2 and b2. Given that a and b are…

Q: For any integer a, prove that a=0 or 1 (mod 3).

A: The solution is given as follows

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

A: To find all integers such that x^3-3 is divisible by x-3

Q: For any non negative integer n , prove that an − bn is divisible by a−b,where a and b are any…

A: answer

Q: •Prove: For all integers a, b, and c, if a|b and alc, then a|(b+ c). we can assume a #0.)

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Q: Prove that for every positive integer m, there is a positive integer k such that k does not divide m…

A: Given:-                 prove that every positive integer m, there is a positive integer K such that…

Q: Suppose that k and n > k are fixed positive integer. Justify the identity n k +1 j=k

A: Recall that KCk+1=0. We will use it in the proof.

Q: Use a direct proof technique to prove the following theorems: For all integers r and y, r + y - 3 is…

A: We have to prove that , For all integers x and y , x2+y2-3 is not divisible by 4. We will have 3…

Q: For each positive integer n, find with proof GCD(220° - 220, 220" + 218).

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

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Q: For an arbitrary integer a, verify the following: (a) 2|a(a + 1), and 3| a(a + 1)(a + 2).

A: We have to verify, 2|a(a+1)  and 3|a(a+1)(a+2) Here we use division algorithm.

Q: Q- Prove the following identities. (a):- 1-0- AV

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Q: Let a and b be two integers. Prove that if a+b is even then a-b is even.

A: Given that a and b are any two integers. We are trying to prove "if a+b is even then a-b is even."

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

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Q: Let a and b be positive integers such that ax + by = 7 for some integers x and y. Then %3D

A: We have to choose  current answer of given statement.

Q: Show that 3·4n+ 51 is divisible by 3 and 9 for all positive integersn

A: Consider the provided expression,

Q: Find a counterexample to show that the statement is Falso- Q. (x+)(R-1) のFor all numlbers X.-メくメ %3D…

A: Part (a) -x<xTake x=-1--1≮-11≮-1 Therefore, the given statement is not true for all values of x.

Q: For an arbitrary integer, say A, verify that 2|A(A+1). Similarly, verify that 3|A(A+1)(A+2).

A: To verify:A(A+1) is divisible by 2.

Q: Suppose that k and n > k are fixed positive integer. Justify the identity n n + 1 %3D k +1, j=k

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Q: Prove, directly from the definition of divisibility, that for any integer x#0, x|0, 1|x and x|x.

A: Use the following definition of divisibility to prove the required result. Let a and b be any two…

Q: Show that for all integers a, b > 0, the number (a + b) – (a O b) is both nonnegative and even.

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

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Q: Use a direct proof technique to prove the following theorems: For all integers z and y, r² + y - 3…

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Q: Let x be an integer. Prove or provide a counterexample: If 4x+7 is odd, then x is odd.

A: We provide a direct proof and give counter example to show that given statement is false.

Q: For every positive integer a, find a composite number n such that n|a" – a.

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Q: Prove: If a is a positive even integer and b is an odd integer, then (a,b)=(a/2, b).

A: Let d be gcd of a and b. Now, b is odd so d cannot be even or is odd.  Now, since d is odd and…

Q: Let x be positive integer. Prove If x2 + 1 is even, then x3 is odd

A: Consider the provided question, Let x be positive integer. We need to Prove that, If x2+1 is even,…

Q: Show for all integers n2 0 that 91 (4" - 1). a.

A: A. We will solve the question using the method of mathematical induction.

Q: Suppose a, b, and c are integers and suppose a b and ałc, where atc denotes that c is not divisible…

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Q: Use the {\bf Cayley Hamilton Theorem} to come up with an expression for A" in terms of n when 1 A =

A: As you have asked more than one question, we shall answer first Que only. For others kindly post…

Q: Show that any integer of the form 6k + 5 is also of the form 3j + 2, but not conversely.

A: 2. We can write 6k+5 = 6k+3+2=3(2k+1)+2 lets write 2k+1=j hence 6k+5=3j+2

Q: Prove that for a positive integer n we have that Bn+3 - 26n – 27 is divisible by 169.

A:

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

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Q: division of a2 with 4 is either 1 or 0.

A: We divide integers into 4 different class. Then work on each class of numbers.

Q: a2 − b2is divisible by 5.

A: First observe that one's place of the square of a positive number can only one among 0,1,4,9,6, and…