Let n > 1 be an integer. We say that [x] ∈ℤ/n is a zero divisor if there exists [y] ∈ ℤ/n − {[0]} such that [x][y] = [0]. We say that [x] is a unit if there exists [y] ∈ ℤ/n such that [x][y] = [1]. 7a) What are all the units in ℤ/12? What are all the zero divisors? b) What are all the units in ℤ/14? What are all the zero divisors? c) What are all the units in ℤ/13? What are all the zero divisors?

Q: 2.47. Let P(x) be the assertion "x is odd", and let Q(x) be the assertion "x2 - 1 is divisible by…

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

Q: Let a, b, x, y ∈Zbe such that ax+by= 11. May we conclude that gcd(a, b) = 11?Why or why not? (Write…

A: Consider the given information:Let a,b,x,y∈ℤ be such that ax+ by=11.

Q: Determine which of these relations are transitive. The variables x, y, x', y represent integers. A.…

A: Note: As multiple questions have been posted, so only A is solved according to QnA guidelines.…

Q: Let H = {0, +-4, +-8, +-12,....}. How many distinct left cosets of H in Z are there? Explain your…

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Q: Prove that Z3 x Zz is isomorphic to Zg.

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Q: 3. Let R-Z18. Determine the units and zero divisors of R giving the explicit unit or zero divisor…

A: We know that  no of units in ring Zn = phi(n), and  no of zero divisor of Zn = n - phi(n) -1

Q: [Formal Proof] Let x in Z. Prove that 7x + 3 is even if and only if 3x - 4 is odd.

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Q: Prove that ∀a, b, k ∈ Z that ((4|k) ∧ (k = ab)) ⇒ (2|a ∨ 2|b)

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Q: Let n and k be integers such that (n, k) = 1. Prove that (n, n − k) = 1.

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Q: Prove that gcd(a, b) is the least natural number representable in the form ax + by with x, y ∈ Z.

A: We have to show that gcd(a, b) is the least natural number representable in the form ax + by with x,…

Q: For allne z, gcdin,nt)=l.

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Q: 9. Show that if a is a root of X²-bX+c with b, c integers and 6² - 4c < 0, then there are only…

A: Given that α is a root of X2-bx+c, with b,c integers and b2-4c&lt;0. Since b2-4c&lt;0, therefore…

Q: {(a, b) ab >0} | {(a, {(a,b) | a² > b} ¯ {(a,b) | a² = b² (mod 7)} b)|a>b-1}

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

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Q: Let z be a root of z5 – 1 = 0 with z # 1. Compute %3D 215 + 216 + z17 +· . + z50

A: Since z is a root of z5-1=0 ⇒z-11+z+z2+z3+z4=0⇒1+z+z2+z3+z4=0…

Q: (1) if is reflexive, symmetric, or transitive. For a, b = Z, define ab if a + b is a multiple of 3.…

A: A relation is a subset of the cartesian product A×B where A and B are non-empty sets. For cartesian…

Q: Let a and b be integers which are not both zero, and let d= gcd(a, b). Prove that gcd (a/d,b/d)= 1.

A: Given that, a and b be integers and in both of them at least one is non-zero. And, greatest common…