In the group of integers Z with addition, cl(5) the conjugation class of 5 is equal to:

Q: Show that for every integer n, n³ + 2 is not divisible by 4.

A: Here, we use rules of divisibility.

Q: Consider the congruence classes Z/18Z: (A) Identify the set S of all zero divisors and the set U of…

A: A) Z/18Z units is given as [n] such that (n,18)=1 , 1≤n≤18⇒n=1,5,7,11,13,17Implies zero divisors ,…

Q: In Z12, the elements [3] and [9] are * Associates Units Irreducibles None of the choices

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Q: The following is a valid identity, true or false? P(k): 1+2+22 +23 + .. +2k = (2(k+1)/2) +…

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Q: Explain why the correspondence x → 3x from Z12 to Z10 is not ahomomorphism.

A: We have to explain that: Why the correspondence x→3x from Z12 to Z10 is not a homomorphism.

Q: Determine all integers n > 1 for which (n-1)! is a zero-divisor in Zn.

A: Given: Zn. To Do:We have to determine all integers n>1 for which n-1! is a zero divisor in Zn.

Q: For each integer n > 1, give examples of two nonisomorphic groupsof order n2.

A: To Do: For each integer n > 1, give examples of two non-isomorphic groups of order n2.

Q: a

A:

Q: 5. Let r,ne N. Show that the product (n+1)(n+2)...(n+r) of r consecutive positive integers is…

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Q: Show that every element of GF(pn) can be written in the form ap forsome unique a in GF( pn).

A: In this question, we have to show that, every element of GF(pn) can be written in the form ap for…

Q: Let a and b be positive integers. Suppose Q is defined recursively as follows: if a < b Q(a, b) =…

A: Given Qa,b=0   if a&lt;bQa-b,b+1  if b≤a.

Q: In Z[2√7], show that N(6 + 2√-7) 5 N(1 + 3√-7) but 6 + 2√-7 and 1 + 3√-7 are not associates.

A: As per the question we are given the extended field ℤ [2√-7] and we have to prove that N(6 + 2√-7) =…

Q: Showthat for any 20 distinct elements of [150], there exists distinct elements a, b, c, d such that…

A: Given: It includes 20 distinct elements of [150] where the elements a, b, c, d such that a + b = c +…

Q: Z12 The order of the element 5+ (4) in the factor group is (4) a) 12 b) 5 c) 4 d) 0

A:

Q: Determine the orders of the elements of D∞

A: It is known that, D∞=a,b|a2=b2=e also it can be observed that, D∞=abka,abk,bak,bakb Let x∈D∞. If…

Q: In the group (Z10, +) (set of integers modulo 10 under addition), compute for (a) 9 + 8 and (b)…

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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 x € Z. Prove that if x² - 8x + 11 is divisible by 4, the x6- 1 is also divisible by 4.

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Q: Find ntegers G and t such that gcd (963,652)3967+t-652,

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Q: Prove that n2+n  is  divisible by 2, where nN.

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Q: Negate the statement: For all positive integers, n, there exists a positive integer, k, so that…

A: The negation of a statement is:- Statement:- p implies q. Negation:- p does not imply q.

Q: Exercise 3. Consider the structure (Z+, *) of the positive integers with multipli- cation. Show that…

A: We will first prove that any set A is definable on the structure (Z+,*) iff : Either P⊆ A  or,  A∩ P…

Q: Disprove the statement: "For all integers r, if r> 2 thenr² - 4 is composite."

A: Take r=3

Q: Find ²** (3z + 2y) ºdzdy

A:

Q: Consider splitting a circle into n sectors. There are k >= 3 colors. We define an as the number of…

A: A recursive relation is a mathematical relationship or formula that expresses a term in a sequence…

Q: Disprove this statement by giving a counterexample. For all nonzero real numbers, a, b, c, and d,…

A: Here a,b,c,d all are non zero real .So (a/b) and (c/d) are defined.  Let,  a = 1 , b = 2 , c = 3 , d…