A multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m.
1. [tex]Z_{11}=\{\overline{0},\overline{1},\overline{2},\overline{3},\overline{4},\overline{5},\overline{6},\overline{7},\overline{8},\overline{9},\overline{10}\}.[/tex]
Check:
- [tex]5\cdot 0=\overline{0};[/tex]
- [tex]5\cdot 1=\overline{5};[/tex]
- [tex]5\cdot 2=\overline{10};[/tex]
- [tex]5\cdot 3=15=\overline{4};[/tex]
- [tex]5\cdot 4=20=\overline{9};[/tex]
- [tex]5\cdot 5=25=\overline{3};[/tex]
- [tex]5\cdot 6=30=\overline{8};[/tex]
- [tex]5\cdot 7=35=\overline{2};[/tex]
- [tex]5\cdot 8=40=\overline{7};[/tex]
- [tex]5\cdot 9=45=\overline{1};[/tex]
- [tex]5\cdot 10=50=\overline{6}.[/tex]
The multiplicative inverse of 5 in [tex]Z_{11}[/tex] is 9.
2. [tex]Z_{12}=\{\overline{0},\overline{1},\overline{2},\overline{3},\overline{4},\overline{5},\overline{6},\overline{7},\overline{8},\overline{9},\overline{10},\overline{11}\}.[/tex]
Check:
- [tex]5\cdot 0=\overline{0};[/tex]
- [tex]5\cdot 1=\overline{5};[/tex]
- [tex]5\cdot 2=\overline{10};[/tex]
- [tex]5\cdot 3=15=\overline{3};[/tex]
- [tex]5\cdot 4=20=\overline{8};[/tex]
- [tex]5\cdot 5=25=\overline{1};[/tex]
- [tex]5\cdot 6=30=\overline{6};[/tex]
- [tex]5\cdot 7=35=\overline{11};[/tex]
- [tex]5\cdot 8=40=\overline{4};[/tex]
- [tex]5\cdot 9=45=\overline{9};[/tex]
- [tex]5\cdot 10=50=\overline{2};[/tex]
- [tex]5\cdot 11=55=\overline{7}.[/tex]
The multiplicative inverse of 5 in [tex]Z_{12}[/tex] is 5.
3. [tex]Z_{13}=\{\overline{0},\overline{1},\overline{2},\overline{3},\overline{4},\overline{5},\overline{6},\overline{7},\overline{8},\overline{9},\overline{10},\overline{11},\overline{12}\}.[/tex]
Check:
- [tex]5\cdot 0=\overline{0};[/tex]
- [tex]5\cdot 1=\overline{5};[/tex]
- [tex]5\cdot 2=\overline{10};[/tex]
- [tex]5\cdot 3=15=\overline{2};[/tex]
- [tex]5\cdot 4=20=\overline{7};[/tex]
- [tex]5\cdot 5=25=\overline{12};[/tex]
- [tex]5\cdot 6=30=\overline{4};[/tex]
- [tex]5\cdot 7=35=\overline{9};[/tex]
- [tex]5\cdot 8=40=\overline{1};[/tex]
- [tex]5\cdot 9=45=\overline{6};[/tex]
- [tex]5\cdot 10=50=\overline{11};[/tex]
- [tex]5\cdot 11=55=\overline{3};[/tex]
- [tex]5\cdot 12=60=\overline{8}.[/tex]
The multiplicative inverse of 5 in [tex]Z_{13}[/tex] is 8.
