Find the order of each of the following elements 5 elementof

Find the order of each of the following elements. 5 elementof Z_12 Squareroot 3 elementof R Squareroot 3 elementof R* -i elementof C* 72 in Z_240 312 in Z_471

Solution

a) The gcd (5,12) is 1. Therefore 1 <5>, so the order of 5 is 12/1 = 12.

b) Infinite

c) Infinite

d) -i can be {1,-1,i,-i}. So the order will be 4.

e) The gcd (72,240) is 24. Therefore 24 <72>, so the order of 72 is 240/24 = 10.

f) The gcd (312, 471) is 3. Therefore 3 <312>, so the order of 312 is 471/3 = 157