the order of element 3 in the group ( ℤ 4 , + ) .

Question

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Answer 1

Question

Chapter 6.3, Problem 8E

(a)

To determine

To find: the order of element 3 in the group (ℤ4,+) .

(a)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ4,+) is 4.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ4,+) .

ℤ4={0,1,2,3,4}

Now, for the element 3 of the group (ℤ4,+) .

1(3)=3=3(mod 4)2(3)=6=2(mod 4)3(3)=9=1(mod 4)4(3)=12=0(mod 4)

Thus, the order of the element 3 of the group (ℤ4,+) is 4.

(b)

To determine

To find: the order of element 3 in the group (ℤ5,+) .

(b)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ5,+) is 5.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ5,+) .

ℤ5={0,1,2,3,4,5}

Now, for the element 3 of the group (ℤ5,+) .

1(3)=3=3(mod 5)2(3)=6=1(mod 5)3(3)=9=4(mod 5)4(3)=12=2(mod 5)5(3)=15=0(mod 5)

Thus, the order of the element 3 of the group (ℤ5,+) is 5.

(c)

To determine

To find: the order of element 3 in the group (ℤ6,+) .

(c)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ6,+) is 2.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ6,+) .

ℤ6={0,1,2,3,4,5,6}

Now, for the element 3 of the group (ℤ6,+) .

1(3)=3=3(mod 6)2(3)=6=0(mod 6)3(3)=9=3(mod 6)4(3)=12=0(mod 6)5(3)=15=3(mod 6)6(3)=18=0(mod 6)

Thus, the order of the element 3 of the group (ℤ6,+) is 2.

(d)

To determine

To find: the order of element 3 in the group (ℤ8,+) .

(d)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ8,+) is 8.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ8,+) .

ℤ8={0,1,2,3,4,5,6,7,8}

Now, for the element 3 of the group (ℤ8,+) .

1(3)=3=3(mod 8)2(3)=6=6(mod 8)3(3)=9=1(mod 8)4(3)=12=4(mod 8)5(3)=15=7(mod 8)6(3)=18=2(mod 8)7(3)=21=5(mod 8)8(3)=24=0(mod 8)

Thus, the order of the element 3 of the group (ℤ8,+) is 8.

(e)

To determine

To find: the order of element 3 in the group (ℤ9,+) .

(e)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ9,+) is 3.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ9,+) .

ℤ9={0,1,2,3,4,5,6,7,8,9}

Now, for the element 3 of the group (ℤ9,+) .

1(3)=3=3(mod 9)2(3)=6=6(mod 9)3(3)=9=0(mod 9)4(3)=12=3(mod 9)5(3)=15=6(mod 9)6(3)=18=0(mod 9)7(3)=21=3(mod 9)8(3)=24=6(mod 9)9(3)=27=0(mod 9)

Thus, the order of the element 3 of the group (ℤ9,+) is 3.

(f)

To determine

To find: the order of element 3 in the group (∪5,·) .

(f)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪5,·) is 4.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪5,·) .

∪5={1,2,3,4}

Now, for the element 3 of the group (∪5,·) .

31=3=3(mod 5)32=9=4(mod 5)33=27=2(mod 5)34=81=1(mod 5)

Thus, the order of the element 3 of the group (∪5,·) is 4.

(g)

To determine

To find: the order of element 3 in the group (∪7,·) .

(g)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪7,·) is 6.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪7,·) .

∪7={1,2,3,4,5,6}

Now, for the element 3 of the group (∪7,·) .

31=3=3(mod 7)32=9=2(mod 7)33=27=6(mod 7)34=81=4(mod 7)35=243=5(mod 7)36=729=1(mod 7)

Thus, the order of the element 3 of the group (∪7,·) is 6.

(h)

To determine

To find: the order of element 3 in the group (∪11,·) .

(h)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪11,·) is 5.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪11,·) .

∪11={1,2,3,4,5,6,7,8,9,10}

Now, for the element 3 of the group (∪11,·) .

31=3=3(mod 11)32=9=9(mod 11)33=27=5(mod 11)34=81=4(mod 11)35=243=1(mod 11)36=729=3(mod 11)37=2187=9(mod 11)

Thus, the order of the element 3 of the group (∪11,·) is 5.

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Answer 2

Question

Chapter 6.3, Problem 8E

(a)

To determine

To find: the order of element 3 in the group (ℤ4,+) .

(a)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ4,+) is 4.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ4,+) .

ℤ4={0,1,2,3,4}

Now, for the element 3 of the group (ℤ4,+) .

1(3)=3=3(mod 4)2(3)=6=2(mod 4)3(3)=9=1(mod 4)4(3)=12=0(mod 4)

Thus, the order of the element 3 of the group (ℤ4,+) is 4.

(b)

To determine

To find: the order of element 3 in the group (ℤ5,+) .

(b)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ5,+) is 5.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ5,+) .

ℤ5={0,1,2,3,4,5}

Now, for the element 3 of the group (ℤ5,+) .

1(3)=3=3(mod 5)2(3)=6=1(mod 5)3(3)=9=4(mod 5)4(3)=12=2(mod 5)5(3)=15=0(mod 5)

Thus, the order of the element 3 of the group (ℤ5,+) is 5.

(c)

To determine

To find: the order of element 3 in the group (ℤ6,+) .

(c)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ6,+) is 2.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ6,+) .

ℤ6={0,1,2,3,4,5,6}

Now, for the element 3 of the group (ℤ6,+) .

1(3)=3=3(mod 6)2(3)=6=0(mod 6)3(3)=9=3(mod 6)4(3)=12=0(mod 6)5(3)=15=3(mod 6)6(3)=18=0(mod 6)

Thus, the order of the element 3 of the group (ℤ6,+) is 2.

(d)

To determine

To find: the order of element 3 in the group (ℤ8,+) .

(d)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ8,+) is 8.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ8,+) .

ℤ8={0,1,2,3,4,5,6,7,8}

Now, for the element 3 of the group (ℤ8,+) .

1(3)=3=3(mod 8)2(3)=6=6(mod 8)3(3)=9=1(mod 8)4(3)=12=4(mod 8)5(3)=15=7(mod 8)6(3)=18=2(mod 8)7(3)=21=5(mod 8)8(3)=24=0(mod 8)

Thus, the order of the element 3 of the group (ℤ8,+) is 8.

(e)

To determine

To find: the order of element 3 in the group (ℤ9,+) .

(e)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ9,+) is 3.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ9,+) .

ℤ9={0,1,2,3,4,5,6,7,8,9}

Now, for the element 3 of the group (ℤ9,+) .

1(3)=3=3(mod 9)2(3)=6=6(mod 9)3(3)=9=0(mod 9)4(3)=12=3(mod 9)5(3)=15=6(mod 9)6(3)=18=0(mod 9)7(3)=21=3(mod 9)8(3)=24=6(mod 9)9(3)=27=0(mod 9)

Thus, the order of the element 3 of the group (ℤ9,+) is 3.

(f)

To determine

To find: the order of element 3 in the group (∪5,·) .

(f)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪5,·) is 4.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪5,·) .

∪5={1,2,3,4}

Now, for the element 3 of the group (∪5,·) .

31=3=3(mod 5)32=9=4(mod 5)33=27=2(mod 5)34=81=1(mod 5)

Thus, the order of the element 3 of the group (∪5,·) is 4.

(g)

To determine

To find: the order of element 3 in the group (∪7,·) .

(g)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪7,·) is 6.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪7,·) .

∪7={1,2,3,4,5,6}

Now, for the element 3 of the group (∪7,·) .

31=3=3(mod 7)32=9=2(mod 7)33=27=6(mod 7)34=81=4(mod 7)35=243=5(mod 7)36=729=1(mod 7)

Thus, the order of the element 3 of the group (∪7,·) is 6.

(h)

To determine

To find: the order of element 3 in the group (∪11,·) .

(h)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪11,·) is 5.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪11,·) .

∪11={1,2,3,4,5,6,7,8,9,10}

Now, for the element 3 of the group (∪11,·) .

31=3=3(mod 11)32=9=9(mod 11)33=27=5(mod 11)34=81=4(mod 11)35=243=1(mod 11)36=729=3(mod 11)37=2187=9(mod 11)

Thus, the order of the element 3 of the group (∪11,·) is 5.

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Answer 3

Question

Chapter 6.3, Problem 8E

(a)

To determine

To find: the order of element 3 in the group (ℤ4,+) .

(a)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ4,+) is 4.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ4,+) .

ℤ4={0,1,2,3,4}

Now, for the element 3 of the group (ℤ4,+) .

1(3)=3=3(mod 4)2(3)=6=2(mod 4)3(3)=9=1(mod 4)4(3)=12=0(mod 4)

Thus, the order of the element 3 of the group (ℤ4,+) is 4.

(b)

To determine

To find: the order of element 3 in the group (ℤ5,+) .

(b)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ5,+) is 5.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ5,+) .

ℤ5={0,1,2,3,4,5}

Now, for the element 3 of the group (ℤ5,+) .

1(3)=3=3(mod 5)2(3)=6=1(mod 5)3(3)=9=4(mod 5)4(3)=12=2(mod 5)5(3)=15=0(mod 5)

Thus, the order of the element 3 of the group (ℤ5,+) is 5.

(c)

To determine

To find: the order of element 3 in the group (ℤ6,+) .

(c)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ6,+) is 2.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ6,+) .

ℤ6={0,1,2,3,4,5,6}

Now, for the element 3 of the group (ℤ6,+) .

1(3)=3=3(mod 6)2(3)=6=0(mod 6)3(3)=9=3(mod 6)4(3)=12=0(mod 6)5(3)=15=3(mod 6)6(3)=18=0(mod 6)

Thus, the order of the element 3 of the group (ℤ6,+) is 2.

(d)

To determine

To find: the order of element 3 in the group (ℤ8,+) .

(d)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ8,+) is 8.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ8,+) .

ℤ8={0,1,2,3,4,5,6,7,8}

Now, for the element 3 of the group (ℤ8,+) .

1(3)=3=3(mod 8)2(3)=6=6(mod 8)3(3)=9=1(mod 8)4(3)=12=4(mod 8)5(3)=15=7(mod 8)6(3)=18=2(mod 8)7(3)=21=5(mod 8)8(3)=24=0(mod 8)

Thus, the order of the element 3 of the group (ℤ8,+) is 8.

(e)

To determine

To find: the order of element 3 in the group (ℤ9,+) .

(e)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ9,+) is 3.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ9,+) .

ℤ9={0,1,2,3,4,5,6,7,8,9}

Now, for the element 3 of the group (ℤ9,+) .

1(3)=3=3(mod 9)2(3)=6=6(mod 9)3(3)=9=0(mod 9)4(3)=12=3(mod 9)5(3)=15=6(mod 9)6(3)=18=0(mod 9)7(3)=21=3(mod 9)8(3)=24=6(mod 9)9(3)=27=0(mod 9)

Thus, the order of the element 3 of the group (ℤ9,+) is 3.

(f)

To determine

To find: the order of element 3 in the group (∪5,·) .

(f)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪5,·) is 4.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪5,·) .

∪5={1,2,3,4}

Now, for the element 3 of the group (∪5,·) .

31=3=3(mod 5)32=9=4(mod 5)33=27=2(mod 5)34=81=1(mod 5)

Thus, the order of the element 3 of the group (∪5,·) is 4.

(g)

To determine

To find: the order of element 3 in the group (∪7,·) .

(g)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪7,·) is 6.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪7,·) .

∪7={1,2,3,4,5,6}

Now, for the element 3 of the group (∪7,·) .

31=3=3(mod 7)32=9=2(mod 7)33=27=6(mod 7)34=81=4(mod 7)35=243=5(mod 7)36=729=1(mod 7)

Thus, the order of the element 3 of the group (∪7,·) is 6.

(h)

To determine

To find: the order of element 3 in the group (∪11,·) .

(h)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪11,·) is 5.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪11,·) .

∪11={1,2,3,4,5,6,7,8,9,10}

Now, for the element 3 of the group (∪11,·) .

31=3=3(mod 11)32=9=9(mod 11)33=27=5(mod 11)34=81=4(mod 11)35=243=1(mod 11)36=729=3(mod 11)37=2187=9(mod 11)

Thus, the order of the element 3 of the group (∪11,·) is 5.

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Answer 4

Question

Chapter 6.3, Problem 8E

(a)

To determine

To find: the order of element 3 in the group (ℤ4,+) .

(a)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ4,+) is 4.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ4,+) .

ℤ4={0,1,2,3,4}

Now, for the element 3 of the group (ℤ4,+) .

1(3)=3=3(mod 4)2(3)=6=2(mod 4)3(3)=9=1(mod 4)4(3)=12=0(mod 4)

Thus, the order of the element 3 of the group (ℤ4,+) is 4.

(b)

To determine

To find: the order of element 3 in the group (ℤ5,+) .

(b)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ5,+) is 5.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ5,+) .

ℤ5={0,1,2,3,4,5}

Now, for the element 3 of the group (ℤ5,+) .

1(3)=3=3(mod 5)2(3)=6=1(mod 5)3(3)=9=4(mod 5)4(3)=12=2(mod 5)5(3)=15=0(mod 5)

Thus, the order of the element 3 of the group (ℤ5,+) is 5.

(c)

To determine

To find: the order of element 3 in the group (ℤ6,+) .

(c)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ6,+) is 2.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ6,+) .

ℤ6={0,1,2,3,4,5,6}

Now, for the element 3 of the group (ℤ6,+) .

1(3)=3=3(mod 6)2(3)=6=0(mod 6)3(3)=9=3(mod 6)4(3)=12=0(mod 6)5(3)=15=3(mod 6)6(3)=18=0(mod 6)

Thus, the order of the element 3 of the group (ℤ6,+) is 2.

(d)

To determine

To find: the order of element 3 in the group (ℤ8,+) .

(d)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ8,+) is 8.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ8,+) .

ℤ8={0,1,2,3,4,5,6,7,8}

Now, for the element 3 of the group (ℤ8,+) .

1(3)=3=3(mod 8)2(3)=6=6(mod 8)3(3)=9=1(mod 8)4(3)=12=4(mod 8)5(3)=15=7(mod 8)6(3)=18=2(mod 8)7(3)=21=5(mod 8)8(3)=24=0(mod 8)

Thus, the order of the element 3 of the group (ℤ8,+) is 8.

(e)

To determine

To find: the order of element 3 in the group (ℤ9,+) .

(e)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ9,+) is 3.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ9,+) .

ℤ9={0,1,2,3,4,5,6,7,8,9}

Now, for the element 3 of the group (ℤ9,+) .

1(3)=3=3(mod 9)2(3)=6=6(mod 9)3(3)=9=0(mod 9)4(3)=12=3(mod 9)5(3)=15=6(mod 9)6(3)=18=0(mod 9)7(3)=21=3(mod 9)8(3)=24=6(mod 9)9(3)=27=0(mod 9)

Thus, the order of the element 3 of the group (ℤ9,+) is 3.

(f)

To determine

To find: the order of element 3 in the group (∪5,·) .

(f)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪5,·) is 4.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪5,·) .

∪5={1,2,3,4}

Now, for the element 3 of the group (∪5,·) .

31=3=3(mod 5)32=9=4(mod 5)33=27=2(mod 5)34=81=1(mod 5)

Thus, the order of the element 3 of the group (∪5,·) is 4.

(g)

To determine

To find: the order of element 3 in the group (∪7,·) .

(g)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪7,·) is 6.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪7,·) .

∪7={1,2,3,4,5,6}

Now, for the element 3 of the group (∪7,·) .

31=3=3(mod 7)32=9=2(mod 7)33=27=6(mod 7)34=81=4(mod 7)35=243=5(mod 7)36=729=1(mod 7)

Thus, the order of the element 3 of the group (∪7,·) is 6.

(h)

To determine

To find: the order of element 3 in the group (∪11,·) .

(h)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪11,·) is 5.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪11,·) .

∪11={1,2,3,4,5,6,7,8,9,10}

Now, for the element 3 of the group (∪11,·) .

31=3=3(mod 11)32=9=9(mod 11)33=27=5(mod 11)34=81=4(mod 11)35=243=1(mod 11)36=729=3(mod 11)37=2187=9(mod 11)

Thus, the order of the element 3 of the group (∪11,·) is 5.

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Answer 5

Chapter 6.3, Problem 8E

(a)

To determine

To find: the order of element 3 in the group (ℤ4,+) .

(a)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ4,+) is 4.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ4,+) .

ℤ4={0,1,2,3,4}

Now, for the element 3 of the group (ℤ4,+) .

1(3)=3=3(mod 4)2(3)=6=2(mod 4)3(3)=9=1(mod 4)4(3)=12=0(mod 4)

Thus, the order of the element 3 of the group (ℤ4,+) is 4.

(b)

To determine

To find: the order of element 3 in the group (ℤ5,+) .

(b)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ5,+) is 5.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ5,+) .

ℤ5={0,1,2,3,4,5}

Now, for the element 3 of the group (ℤ5,+) .

1(3)=3=3(mod 5)2(3)=6=1(mod 5)3(3)=9=4(mod 5)4(3)=12=2(mod 5)5(3)=15=0(mod 5)

Thus, the order of the element 3 of the group (ℤ5,+) is 5.

(c)

To determine

To find: the order of element 3 in the group (ℤ6,+) .

(c)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ6,+) is 2.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ6,+) .

ℤ6={0,1,2,3,4,5,6}

Now, for the element 3 of the group (ℤ6,+) .

1(3)=3=3(mod 6)2(3)=6=0(mod 6)3(3)=9=3(mod 6)4(3)=12=0(mod 6)5(3)=15=3(mod 6)6(3)=18=0(mod 6)

Thus, the order of the element 3 of the group (ℤ6,+) is 2.

(d)

To determine

To find: the order of element 3 in the group (ℤ8,+) .

(d)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ8,+) is 8.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ8,+) .

ℤ8={0,1,2,3,4,5,6,7,8}

Now, for the element 3 of the group (ℤ8,+) .

1(3)=3=3(mod 8)2(3)=6=6(mod 8)3(3)=9=1(mod 8)4(3)=12=4(mod 8)5(3)=15=7(mod 8)6(3)=18=2(mod 8)7(3)=21=5(mod 8)8(3)=24=0(mod 8)

Thus, the order of the element 3 of the group (ℤ8,+) is 8.

(e)

To determine

To find: the order of element 3 in the group (ℤ9,+) .

(e)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ9,+) is 3.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ9,+) .

ℤ9={0,1,2,3,4,5,6,7,8,9}

Now, for the element 3 of the group (ℤ9,+) .

1(3)=3=3(mod 9)2(3)=6=6(mod 9)3(3)=9=0(mod 9)4(3)=12=3(mod 9)5(3)=15=6(mod 9)6(3)=18=0(mod 9)7(3)=21=3(mod 9)8(3)=24=6(mod 9)9(3)=27=0(mod 9)

Thus, the order of the element 3 of the group (ℤ9,+) is 3.

(f)

To determine

To find: the order of element 3 in the group (∪5,·) .

(f)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪5,·) is 4.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪5,·) .

∪5={1,2,3,4}

Now, for the element 3 of the group (∪5,·) .

31=3=3(mod 5)32=9=4(mod 5)33=27=2(mod 5)34=81=1(mod 5)

Thus, the order of the element 3 of the group (∪5,·) is 4.

(g)

To determine

To find: the order of element 3 in the group (∪7,·) .

(g)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪7,·) is 6.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪7,·) .

∪7={1,2,3,4,5,6}

Now, for the element 3 of the group (∪7,·) .

31=3=3(mod 7)32=9=2(mod 7)33=27=6(mod 7)34=81=4(mod 7)35=243=5(mod 7)36=729=1(mod 7)

Thus, the order of the element 3 of the group (∪7,·) is 6.

(h)

To determine

To find: the order of element 3 in the group (∪11,·) .

(h)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪11,·) is 5.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪11,·) .

∪11={1,2,3,4,5,6,7,8,9,10}

Now, for the element 3 of the group (∪11,·) .

31=3=3(mod 11)32=9=9(mod 11)33=27=5(mod 11)34=81=4(mod 11)35=243=1(mod 11)36=729=3(mod 11)37=2187=9(mod 11)

Thus, the order of the element 3 of the group (∪11,·) is 5.

Answer 6

Chapter 6.3, Problem 8E

(a)

To determine

To find: the order of element 3 in the group (ℤ4,+) .

(a)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ4,+) is 4.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ4,+) .

ℤ4={0,1,2,3,4}

Now, for the element 3 of the group (ℤ4,+) .

1(3)=3=3(mod 4)2(3)=6=2(mod 4)3(3)=9=1(mod 4)4(3)=12=0(mod 4)

Thus, the order of the element 3 of the group (ℤ4,+) is 4.

Answer 7

Chapter 6.3, Problem 8E

(a)

To determine

To find: the order of element 3 in the group (ℤ4,+) .

Answer 8

(a)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ4,+) is 4.

Answer 9

(a)

Expert Solution

Answer 10

(a)

Expert Solution

Answer 11

Expert Solution

Answer 12

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ4,+) .

ℤ4={0,1,2,3,4}

Now, for the element 3 of the group (ℤ4,+) .

1(3)=3=3(mod 4)2(3)=6=2(mod 4)3(3)=9=1(mod 4)4(3)=12=0(mod 4)

Thus, the order of the element 3 of the group (ℤ4,+) is 4.

Answer 13

(b)

To determine

To find: the order of element 3 in the group (ℤ5,+) .

(b)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ5,+) is 5.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ5,+) .

ℤ5={0,1,2,3,4,5}

Now, for the element 3 of the group (ℤ5,+) .

1(3)=3=3(mod 5)2(3)=6=1(mod 5)3(3)=9=4(mod 5)4(3)=12=2(mod 5)5(3)=15=0(mod 5)

Thus, the order of the element 3 of the group (ℤ5,+) is 5.

Answer 14

(b)

To determine

To find: the order of element 3 in the group (ℤ5,+) .

Answer 15

(b)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ5,+) is 5.

Answer 16

(b)

Expert Solution

Answer 17

(b)

Expert Solution

Answer 18

Expert Solution

Answer 19

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ5,+) .

ℤ5={0,1,2,3,4,5}

Now, for the element 3 of the group (ℤ5,+) .

1(3)=3=3(mod 5)2(3)=6=1(mod 5)3(3)=9=4(mod 5)4(3)=12=2(mod 5)5(3)=15=0(mod 5)

Thus, the order of the element 3 of the group (ℤ5,+) is 5.

Answer 20

(c)

To determine

To find: the order of element 3 in the group (ℤ6,+) .

(c)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ6,+) is 2.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ6,+) .

ℤ6={0,1,2,3,4,5,6}

Now, for the element 3 of the group (ℤ6,+) .

1(3)=3=3(mod 6)2(3)=6=0(mod 6)3(3)=9=3(mod 6)4(3)=12=0(mod 6)5(3)=15=3(mod 6)6(3)=18=0(mod 6)

Thus, the order of the element 3 of the group (ℤ6,+) is 2.

Answer 21

(c)

To determine

To find: the order of element 3 in the group (ℤ6,+) .

Answer 22

(c)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ6,+) is 2.

Answer 23

(c)

Expert Solution

Answer 24

(c)

Expert Solution

Answer 25

Expert Solution

Answer 26

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ6,+) .

ℤ6={0,1,2,3,4,5,6}

Now, for the element 3 of the group (ℤ6,+) .

1(3)=3=3(mod 6)2(3)=6=0(mod 6)3(3)=9=3(mod 6)4(3)=12=0(mod 6)5(3)=15=3(mod 6)6(3)=18=0(mod 6)

Thus, the order of the element 3 of the group (ℤ6,+) is 2.

Answer 27

(d)

To determine

To find: the order of element 3 in the group (ℤ8,+) .

(d)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ8,+) is 8.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ8,+) .

ℤ8={0,1,2,3,4,5,6,7,8}

Now, for the element 3 of the group (ℤ8,+) .

1(3)=3=3(mod 8)2(3)=6=6(mod 8)3(3)=9=1(mod 8)4(3)=12=4(mod 8)5(3)=15=7(mod 8)6(3)=18=2(mod 8)7(3)=21=5(mod 8)8(3)=24=0(mod 8)

Thus, the order of the element 3 of the group (ℤ8,+) is 8.

Answer 28

(d)

To determine

To find: the order of element 3 in the group (ℤ8,+) .

Answer 29

(d)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ8,+) is 8.

Answer 30

(d)

Expert Solution

Answer 31

(d)

Expert Solution

Answer 32

Expert Solution

Answer 33

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ8,+) .

ℤ8={0,1,2,3,4,5,6,7,8}

Now, for the element 3 of the group (ℤ8,+) .

1(3)=3=3(mod 8)2(3)=6=6(mod 8)3(3)=9=1(mod 8)4(3)=12=4(mod 8)5(3)=15=7(mod 8)6(3)=18=2(mod 8)7(3)=21=5(mod 8)8(3)=24=0(mod 8)

Thus, the order of the element 3 of the group (ℤ8,+) is 8.

Answer 34

(e)

To determine

To find: the order of element 3 in the group (ℤ9,+) .

(e)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ9,+) is 3.

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ9,+) .

ℤ9={0,1,2,3,4,5,6,7,8,9}

Now, for the element 3 of the group (ℤ9,+) .

1(3)=3=3(mod 9)2(3)=6=6(mod 9)3(3)=9=0(mod 9)4(3)=12=3(mod 9)5(3)=15=6(mod 9)6(3)=18=0(mod 9)7(3)=21=3(mod 9)8(3)=24=6(mod 9)9(3)=27=0(mod 9)

Thus, the order of the element 3 of the group (ℤ9,+) is 3.

Answer 35

(e)

To determine

To find: the order of element 3 in the group (ℤ9,+) .

Answer 36

(e)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (ℤ9,+) is 3.

Answer 37

(e)

Expert Solution

Answer 38

(e)

Expert Solution

Answer 39

Expert Solution

Answer 40

Explanation of Solution

Concept used:

The order of an element ‘a’ of an additive group is defined as the least positive integer n such that na=0 , where 0 is the identity element in the additive group.

Calculation:

Consider the group (ℤ9,+) .

ℤ9={0,1,2,3,4,5,6,7,8,9}

Now, for the element 3 of the group (ℤ9,+) .

1(3)=3=3(mod 9)2(3)=6=6(mod 9)3(3)=9=0(mod 9)4(3)=12=3(mod 9)5(3)=15=6(mod 9)6(3)=18=0(mod 9)7(3)=21=3(mod 9)8(3)=24=6(mod 9)9(3)=27=0(mod 9)

Thus, the order of the element 3 of the group (ℤ9,+) is 3.

Answer 41

(f)

To determine

To find: the order of element 3 in the group (∪5,·) .

(f)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪5,·) is 4.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪5,·) .

∪5={1,2,3,4}

Now, for the element 3 of the group (∪5,·) .

31=3=3(mod 5)32=9=4(mod 5)33=27=2(mod 5)34=81=1(mod 5)

Thus, the order of the element 3 of the group (∪5,·) is 4.

Answer 42

(f)

To determine

To find: the order of element 3 in the group (∪5,·) .

Answer 43

(f)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪5,·) is 4.

Answer 44

(f)

Expert Solution

Answer 45

(f)

Expert Solution

Answer 46

Expert Solution

Answer 47

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪5,·) .

∪5={1,2,3,4}

Now, for the element 3 of the group (∪5,·) .

31=3=3(mod 5)32=9=4(mod 5)33=27=2(mod 5)34=81=1(mod 5)

Thus, the order of the element 3 of the group (∪5,·) is 4.

Answer 48

(g)

To determine

To find: the order of element 3 in the group (∪7,·) .

(g)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪7,·) is 6.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪7,·) .

∪7={1,2,3,4,5,6}

Now, for the element 3 of the group (∪7,·) .

31=3=3(mod 7)32=9=2(mod 7)33=27=6(mod 7)34=81=4(mod 7)35=243=5(mod 7)36=729=1(mod 7)

Thus, the order of the element 3 of the group (∪7,·) is 6.

Answer 49

(g)

To determine

To find: the order of element 3 in the group (∪7,·) .

Answer 50

(g)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪7,·) is 6.

Answer 51

(g)

Expert Solution

Answer 52

(g)

Expert Solution

Answer 53

Expert Solution

Answer 54

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪7,·) .

∪7={1,2,3,4,5,6}

Now, for the element 3 of the group (∪7,·) .

31=3=3(mod 7)32=9=2(mod 7)33=27=6(mod 7)34=81=4(mod 7)35=243=5(mod 7)36=729=1(mod 7)

Thus, the order of the element 3 of the group (∪7,·) is 6.

Answer 55

(h)

To determine

To find: the order of element 3 in the group (∪11,·) .

(h)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪11,·) is 5.

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪11,·) .

∪11={1,2,3,4,5,6,7,8,9,10}

Now, for the element 3 of the group (∪11,·) .

31=3=3(mod 11)32=9=9(mod 11)33=27=5(mod 11)34=81=4(mod 11)35=243=1(mod 11)36=729=3(mod 11)37=2187=9(mod 11)

Thus, the order of the element 3 of the group (∪11,·) is 5.

Answer 56

(h)

To determine

To find: the order of element 3 in the group (∪11,·) .

Answer 57

(h)

Expert Solution

Answer to Problem 8E

The order of the element 3 of the group (∪11,·) is 5.

Answer 58

(h)

Expert Solution

Answer 59

(h)

Expert Solution

Answer 60

Expert Solution

Answer 61

Explanation of Solution

Concept used:

The order of an element ‘a’ of a multiplicative group is defined as the least positive integer n such that an=1 , where 1 is the identity element in the multiplicative group.

Calculation:

Consider the group (∪11,·) .

∪11={1,2,3,4,5,6,7,8,9,10}