Mathematics · Advanced university
Order of an element in an additive cyclic group
The operation is addition modulo n. Element order is the least positive multiple giving zero.
Method and conditions
The operation is addition modulo n. Element order is the least positive multiple giving zero.
ord([a] in Z/nZ) = n/gcd(a,n)
Worked examples
In the additive group Z/41Z, what is the order of the residue class [20]? Round the final result to 4 decimal places if needed.
We need the least positive k with 41 dividing 20k. Divide 41 by gcd(20,41) = 1.
41
In the additive group Z/43Z, what is the order of the residue class [8]? Round the final result to 4 decimal places if needed.
We need the least positive k with 43 dividing 8k. Divide 43 by gcd(8,43) = 1.
43
Try it
In the additive group Z/41Z, what is the order of the residue class [35]? Round the final result to 4 decimal places if needed.
In the additive group Z/53Z, what is the order of the residue class [42]? Round the final result to 4 decimal places if needed.
Check your work
Answers and marking notes
- 41
We need the least positive k with 41 dividing 35k. Divide 41 by gcd(35,41) = 1.
- 53
We need the least positive k with 53 dividing 42k. Divide 53 by gcd(42,53) = 1.
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