in the ring a such that (a)=12) +<18) of the integers, Z, find a positile integer

Q: Prove that Z ∼= nZ, for every nonzero integer n

A: Before we start the proof, let's define the terms:Isomorphism: In mathematics, an isomorphism is a…

Q: show that for all integers x and y (x-y)3=x3-y3(mod3)

A:

Q: let a and b be two relatively prime integers. if there exists an m such that (a/b)^m is an integers,…

A:

Q: Let a and b be integers with GCD(a, b) = d and let a = da', b = db'. Prove that GCD(a', b')=1

A: and  be integer such that .It is given that  and .We have to prove .Note: According to the…

Q: Let a be an integer greater than 0 and b be an integer greater than or equal to 0. Let a be greater…

A:

Q: Prove that if D is an integral domain, then either char(D) = 0 or char(D) = p for some prime p. In…

A: Step 1:Definitions and Preliminary ConceptsIntegral Domain: An integral domain DDD is a commutative…

Q: Let [8]131 - [42]131 [al131, where a is a nonnegative integer less than 131. What is a? Answer: 34 X

A: The notation [x]y denotes the remainder when x is divided by y. For example, [8]131 represents the…

Q: 61 Let A = Compute 612 - A and (612)A. 4 2

A: Step 1: Step 2: Step 3: Step 4:

Q: Let f and g be permutations. Show that the order of h=fgf^−1 is the same as the order of g.

A:

Q: Prove that for every positive integer n, Q/Z has an element of order n.

A:

Q: Prove that, IXI<a Rf and If -aL X Z a

A:

Q: Let n be an element in Z. Prove that 5|n^2 if and only if 5|n.

A: If n be an element in Z. We have to prove that 5|n^2 if and only if 5|n.

Q: Let x be an integer. Prove that, if xmod15 = 4, then 15|(x2-1)

A:

Q: Let A be the set of number in (3/5, 2/3) that have decimal expansions containing only finitely many…

A: Consider, Sn=0.66666.....666 ( where 6  appears n times). This is the largest n digit number in the…

Q: a(1–r") for every 1-r Let r + 1 be a real number. Use induction to prove that a + ar + ar² + . ..+…

A:

Q: integer 107] (n²-71) ? Does there exist an such that

A: NO , DOES NOT EXIST SUCH INTEGER n

Q: Prove gcd(n, n+1) = 1.

A: To prove the below expression. gcdn, n+1=1

Q: - If n is a positive composite integer, prove that there exist a, b € Zn such that à 0 and 60 but ab…

A:

Q: Recall that a is a type 1 integer if there exists an integer k such that r = 3k + 1. Prove that if a…

A: x is a type 1 integer. So, x = 3k+1, for some integer k. So, x² = (3k+1)²