Is U18 a cyclic group Is U20 a cyclic groupSolutionU18157111

Is U(18) a cyclic group? Is U(20) a cyclic group?

U18={1,5,7,11,13,17}. Notice 5^2 =25=7 mod18 and 5^3 = 7*5=35=17 mod18. You know the |5| divides |U18|=6. We know that |5| >3, hence it must be 6. So 5 is a generator of U18.

so it is a cyclic group

similery U(20)

U20={1,3,7,9,11,13,17,19}. Notice 3^4=1mod20, 7^4=1mod20, 9^2=1mod20, 11^2=1mod20, 13^4=1mod20, 17^4=1mod20 and 19^2=1mod20. Hence U20 is not cyclic because there is no element of order 8.

U(20)is not a cyclic group

$Is U(18) a cyclic group? Is U(20) a cyclic group?SolutionU18={1,5,7,11,13,17}. Notice 5^2 =25=7 mod18 and 5^3 = 7*5=35=17 mod18. You know the |5| divides |U18|$

Is U18 a cyclic group Is U20 a cyclic groupSolutionU18157111

Solution

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