find the number of soloution to the equation x^9=ein the group z18how many solution are to the equation x^3=e in this group?how many element of order 9 are in this group?hint; for an element to be of order 9 , it must solve x^9=e , and not solve x^n=e for any lower value of n

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Olivia Petersen · Accepted Answer

Clearly, if x is divisible by 2 or 3, x^n is divisible by 2 or 3 for all n, so x is not a solution.There are 6 values remaining, 1, 5, 7, 11, 13, 17If x = 5 mod 6, then x^n = 5 mod 6 for all n oddThus, 5, 11, and 17 are not solutions to either x^3 = e or x ^9 = eThus, we are left with 1, 7, and 13They are all equal to 1 mod 6Thus, for each of them x^3 = 1(To see this, x^3 = (6y+1)^3 = 216y^3 + 108y^2 + 18y^2 + 1 = 1 mod 18 (216, 108, and 18 are all divisible by 18)Then, of course 1^n = 1 for all n, so, as 3 divides 9,1, 7, 13 are solutions of both x^9 = e and x^3 = eThus, there are no elements of order 9.

find the number of soloution to the equation x^9=e in the group z18 how many solution are to the equation x^3=e in this group? how many element of order 9 are in this group? hint; for an element to be of order 9 , it must solve x^9=e , and not solve x^n=e for any lower value of n

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