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Best answer
9 votes
9 votes

$\left ( Z,+_m \right )$ is a standard cyclic group.

And no of generators in a cyclic group is = $\phi(n)$ where n is the order of the group.

$\phi(15) = \phi(3^1 * 5^1) = (3^1-3^0)*(5^1-5^0) = 2* 4 = 8$ generators. 


PS: how to calculate $\phi\left ( n \right ) = \phi\left ( p_1^{x_1}.p_2^{x_2} \right ) = \ \left ( p_1^{x_1} -p_1^{x_1-1} \right ).\left ( p_2^{x_2} -p_2^{x_2-1} \right )$

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4 votes

The number of generators in a group is based on the order of the group.

The no. of generators of a group will always be less than the order of the group and co-prime to the order of the group.

Here O(g) = 15,

Co-primes of 15 are 1,2,4,7,8,11,13,14

Thus the number of generators for this group is 8.

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