Question

What is the multiplicative inverse of 5 in z11, z12, and z13? you can do a trial-and-error search using a calculator or a pc?

Answer 1

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. $Z_{11}=\{\overline{0},\overline{1},\overline{2},\overline{3},\overline{4},\overline{5},\overline{6},\overline{7},\overline{8},\overline{9},\overline{10}\}.$

Check:

• $5\cdot 0=\overline{0};$
• $5\cdot 1=\overline{5};$
• $5\cdot 2=\overline{10};$
• $5\cdot 3=15=\overline{4};$
• $5\cdot 4=20=\overline{9};$
• $5\cdot 5=25=\overline{3};$
• $5\cdot 6=30=\overline{8};$
• $5\cdot 7=35=\overline{2};$
• $5\cdot 8=40=\overline{7};$
• $5\cdot 9=45=\overline{1};$
• $5\cdot 10=50=\overline{6}.$

The multiplicative inverse of 5 in $Z_{11}$ is 9.

2.   $Z_{12}=\{\overline{0},\overline{1},\overline{2},\overline{3},\overline{4},\overline{5},\overline{6},\overline{7},\overline{8},\overline{9},\overline{10},\overline{11}\}.$

Check:

• $5\cdot 0=\overline{0};$
• $5\cdot 1=\overline{5};$
• $5\cdot 2=\overline{10};$
• $5\cdot 3=15=\overline{3};$
• $5\cdot 4=20=\overline{8};$
• $5\cdot 5=25=\overline{1};$
• $5\cdot 6=30=\overline{6};$
• $5\cdot 7=35=\overline{11};$
• $5\cdot 8=40=\overline{4};$
• $5\cdot 9=45=\overline{9};$
• $5\cdot 10=50=\overline{2};$
• $5\cdot 11=55=\overline{7}.$

The multiplicative inverse of 5 in $Z_{12}$ is 5.

3.  $Z_{13}=\{\overline{0},\overline{1},\overline{2},\overline{3},\overline{4},\overline{5},\overline{6},\overline{7},\overline{8},\overline{9},\overline{10},\overline{11},\overline{12}\}.$

Check:

• $5\cdot 0=\overline{0};$
• $5\cdot 1=\overline{5};$
• $5\cdot 2=\overline{10};$
• $5\cdot 3=15=\overline{2};$
• $5\cdot 4=20=\overline{7};$
• $5\cdot 5=25=\overline{12};$
• $5\cdot 6=30=\overline{4};$
• $5\cdot 7=35=\overline{9};$
• $5\cdot 8=40=\overline{1};$
• $5\cdot 9=45=\overline{6};$
• $5\cdot 10=50=\overline{11};$
• $5\cdot 11=55=\overline{3};$
• $5\cdot 12=60=\overline{8}.$

The multiplicative inverse of 5 in $Z_{13}$ is 8.