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How to Find the Largest Order for Any Element in Z15

Z15 is the group defined by the numbers 0 through 14. The operation for the group is “+,” where a + b is defined as the remainder when the sum of a and b is divided by 15. In Z15, 7 + 11 = 3 because 3 is the remainder when the sum of 7 and 11, which is 18. The “order” of an element in Z15, is the minimum number of times the element must be added to itself to equal 0. The order of element x is the smallest (m), such that mx = 0.

Instructions

- 1
  Find the element with the largest order by computing the order of each element. The order of 0 is 1 because the sum of only one 0 is 0. You need 15 1s to add to 0, so 1 is certainly a good candidate for the element with the largest order.
- 2
  Compute the sequence for 2. It is 2, 4, 6, 8, 10 , 12, 14, 1, 3, 5, 7, 9, 11, 13, 0. The order for 2 is also 15. For 3, the sequence is 3, 6, 9, 12, 0. The order of 3 is 5.
- 3
  Look at the sequence for 4. It is 4, 8, 12, 1, 5, 9, 13, 2, 6, 10, 14, 3, 7, 11, 0. The order of 4 is also 15. There is no longest order. At least 1, 2 and 4 have order 15, which is the largest possible order.

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