Find all generators of Z 6 , Z 8 , and Z 20 .

Question

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Answer 1

Textbook Question

Chapter 4, Problem 1E

Find all generators of Z 6 , Z 8 , and Z 20 .

Expert Solution & Answer

To determine

To find:The generators of Z6 ,Z8 and Z20 .

Answer to Problem 1E

The generator of Z6 is {1,5} , generator of Z8 is {1,3,5,7} and Z20 is {1,3,7,9,11,13,17,19}

Explanation of Solution

Given information: (Zn,⊕n) is a group under addition modulo n.

For a group G, if there exist an element a∈G such that G={an:n∈Z} , then such an element a is called generator of the group.

In the group (Zn,⊕n), an integer k is said to be generator of Zn if and only if gcd(k,n)=1 .

For group Z6, the set of integers whose gcd(k,n)=1 is {1,5}

Therefore, in Z6, there are two generators given by {1,5} .

Similarly,

For group Z8, the set of integers whose gcd(k,n)=1 is {1,3,5,7}

Therefore, in Z6, there are four generators given by {1,3,5,7} .

And

For group Z20, the set of integers whose gcd(k,n)=1 is {1,3,7,9,11,13,17,19}

Therefore, in Z6, there are eight generators given by {1,3,7,9,11,13,17,19} .

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Find the perimeter and area

Answer 2

Textbook Question

Chapter 4, Problem 1E

Find all generators of Z 6 , Z 8 , and Z 20 .

Expert Solution & Answer

To determine

To find:The generators of Z6 ,Z8 and Z20 .

Answer to Problem 1E

The generator of Z6 is {1,5} , generator of Z8 is {1,3,5,7} and Z20 is {1,3,7,9,11,13,17,19}

Explanation of Solution

Given information: (Zn,⊕n) is a group under addition modulo n.

For a group G, if there exist an element a∈G such that G={an:n∈Z} , then such an element a is called generator of the group.

In the group (Zn,⊕n), an integer k is said to be generator of Zn if and only if gcd(k,n)=1 .

For group Z6, the set of integers whose gcd(k,n)=1 is {1,5}

Therefore, in Z6, there are two generators given by {1,5} .

Similarly,

For group Z8, the set of integers whose gcd(k,n)=1 is {1,3,5,7}

Therefore, in Z6, there are four generators given by {1,3,5,7} .

And

For group Z20, the set of integers whose gcd(k,n)=1 is {1,3,7,9,11,13,17,19}

Therefore, in Z6, there are eight generators given by {1,3,7,9,11,13,17,19} .

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Answer 3

Textbook Question

Chapter 4, Problem 1E

Find all generators of Z 6 , Z 8 , and Z 20 .

Expert Solution & Answer

To determine

To find:The generators of Z6 ,Z8 and Z20 .

Answer to Problem 1E

The generator of Z6 is {1,5} , generator of Z8 is {1,3,5,7} and Z20 is {1,3,7,9,11,13,17,19}

Explanation of Solution

Given information: (Zn,⊕n) is a group under addition modulo n.

For a group G, if there exist an element a∈G such that G={an:n∈Z} , then such an element a is called generator of the group.

In the group (Zn,⊕n), an integer k is said to be generator of Zn if and only if gcd(k,n)=1 .

For group Z6, the set of integers whose gcd(k,n)=1 is {1,5}

Therefore, in Z6, there are two generators given by {1,5} .

Similarly,

For group Z8, the set of integers whose gcd(k,n)=1 is {1,3,5,7}

Therefore, in Z6, there are four generators given by {1,3,5,7} .

And

For group Z20, the set of integers whose gcd(k,n)=1 is {1,3,7,9,11,13,17,19}

Therefore, in Z6, there are eight generators given by {1,3,7,9,11,13,17,19} .

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Answer 4

Textbook Question

Chapter 4, Problem 1E

Find all generators of Z 6 , Z 8 , and Z 20 .

Expert Solution & Answer

To determine

To find:The generators of Z6 ,Z8 and Z20 .

Answer to Problem 1E

The generator of Z6 is {1,5} , generator of Z8 is {1,3,5,7} and Z20 is {1,3,7,9,11,13,17,19}

Explanation of Solution

Given information: (Zn,⊕n) is a group under addition modulo n.

For a group G, if there exist an element a∈G such that G={an:n∈Z} , then such an element a is called generator of the group.

In the group (Zn,⊕n), an integer k is said to be generator of Zn if and only if gcd(k,n)=1 .

For group Z6, the set of integers whose gcd(k,n)=1 is {1,5}

Therefore, in Z6, there are two generators given by {1,5} .

Similarly,

For group Z8, the set of integers whose gcd(k,n)=1 is {1,3,5,7}

Therefore, in Z6, there are four generators given by {1,3,5,7} .

And

For group Z20, the set of integers whose gcd(k,n)=1 is {1,3,7,9,11,13,17,19}

Therefore, in Z6, there are eight generators given by {1,3,7,9,11,13,17,19} .

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Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

Expert Solution & Answer

To determine

To find:The generators of Z6 ,Z8 and Z20 .

Answer to Problem 1E

The generator of Z6 is {1,5} , generator of Z8 is {1,3,5,7} and Z20 is {1,3,7,9,11,13,17,19}

Explanation of Solution

Given information: (Zn,⊕n) is a group under addition modulo n.

For a group G, if there exist an element a∈G such that G={an:n∈Z} , then such an element a is called generator of the group.

In the group (Zn,⊕n), an integer k is said to be generator of Zn if and only if gcd(k,n)=1 .

For group Z6, the set of integers whose gcd(k,n)=1 is {1,5}

Therefore, in Z6, there are two generators given by {1,5} .

Similarly,

For group Z8, the set of integers whose gcd(k,n)=1 is {1,3,5,7}

Therefore, in Z6, there are four generators given by {1,3,5,7} .

And

For group Z20, the set of integers whose gcd(k,n)=1 is {1,3,7,9,11,13,17,19}

Therefore, in Z6, there are eight generators given by {1,3,7,9,11,13,17,19} .

Answer 8

Expert Solution & Answer

To determine

To find:The generators of Z6 ,Z8 and Z20 .

Answer to Problem 1E

The generator of Z6 is {1,5} , generator of Z8 is {1,3,5,7} and Z20 is {1,3,7,9,11,13,17,19}

Explanation of Solution

Given information: (Zn,⊕n) is a group under addition modulo n.

For a group G, if there exist an element a∈G such that G={an:n∈Z} , then such an element a is called generator of the group.

In the group (Zn,⊕n), an integer k is said to be generator of Zn if and only if gcd(k,n)=1 .

For group Z6, the set of integers whose gcd(k,n)=1 is {1,5}

Therefore, in Z6, there are two generators given by {1,5} .

Similarly,

For group Z8, the set of integers whose gcd(k,n)=1 is {1,3,5,7}

Therefore, in Z6, there are four generators given by {1,3,5,7} .

And

For group Z20, the set of integers whose gcd(k,n)=1 is {1,3,7,9,11,13,17,19}

Therefore, in Z6, there are eight generators given by {1,3,7,9,11,13,17,19} .

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

To determine

To find:The generators of Z6 ,Z8 and Z20 .

Answer 12

Answer to Problem 1E

The generator of Z6 is {1,5} , generator of Z8 is {1,3,5,7} and Z20 is {1,3,7,9,11,13,17,19}

Answer 13

Explanation of Solution

Given information: (Zn,⊕n) is a group under addition modulo n.

For a group G, if there exist an element a∈G such that G={an:n∈Z} , then such an element a is called generator of the group.

In the group (Zn,⊕n), an integer k is said to be generator of Zn if and only if gcd(k,n)=1 .

For group Z6, the set of integers whose gcd(k,n)=1 is {1,5}

Therefore, in Z6, there are two generators given by {1,5} .

Similarly,

For group Z8, the set of integers whose gcd(k,n)=1 is {1,3,5,7}

Therefore, in Z6, there are four generators given by {1,3,5,7} .

And

For group Z20, the set of integers whose gcd(k,n)=1 is {1,3,7,9,11,13,17,19}

Therefore, in Z6, there are eight generators given by {1,3,7,9,11,13,17,19} .

Answer 14

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Find all generators of Z 6 , Z 8 , and Z 20 .

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Answer to Problem 1E

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