Counting Elements of a Given Order

Question

How many elements of the additive group $\mathbb{Z}_{36}$ have order exactly $6$ ?

a. $1$
b. $2$
c. $6$
d. $12$

Correct answer

B. $2$

Full reasoning

1In $\mathbb{Z}_{n}$ the element $k$ has order $n/\gcd(k,n)$ , so here the order of $k$ is $36/\gcd(k,36)$ .
2Setting $36/\gcd(k,36) = 6$ gives $\gcd(k,36) = 6$ : the search is for the $k$ in $\{0,1,\dots,35\}$ whose greatest common divisor with $36$ is exactly $6$ .
3Any such $k$ is a multiple of $6$ , so $k \in \{0, 6, 12, 18, 24, 30\}$ . Their gcds with $36$ are $36, 6, 12, 18, 12, 6$ respectively.
4Only $k = 6$ and $k = 30$ give $\gcd = 6$ , so exactly $2$ elements have order $6$ .
5Cross-check with the general rule: a cyclic group of order $n$ has $\varphi(d)$ elements of order $d$ for each divisor $d$ of $n$ , and $\varphi(6) = 2$ , matching the direct count.

Why each choice is right or wrong

Choice A

This confuses elements with subgroups. $\mathbb{Z}_{36}$ has exactly one subgroup of order $6$ , but that subgroup contains several elements, and only its generators have order $6$ .

Choice B

Correct. An element $k$ has order $36/\gcd(k,36) = 6$ exactly when $\gcd(k,36) = 6$ , which holds only for $k = 6$ and $k = 30$ .

Choice C

This counts every element of the unique order- $6$ subgroup $\{0, 6, 12, 18, 24, 30\}$ . Four of those have smaller order — for example $12$ has order $3$ and $18$ has order $2$ — so they are not elements of order $6$ .

Choice D

This is $\varphi(36) = 12$ , the number of generators of the whole group, that is, the count of elements of order $36$ . The totient must be evaluated at the target order $6$ , not at the group order.

Related formula

Order of an element of a cyclic group

\left| k \right|_{\mathbb{Z}_n} = \frac{n}{\gcd(k, n)}

Assumptions: The group is cyclic of finite order n. The number of elements of order d is phi(d) for each divisor d of n, and zero otherwise.

Sources

Authored for this public explainer registry. It has no database question ID and is not copied from a protected or official exam bank.

Abstract Algebra: Theory and Applications (Judson), Section 4.1: Cyclic Subgroups — LibreTexts Mathematics. Accessed 2026-08-15. Judson's Abstract Algebra on LibreTexts is released under the GNU Free Documentation License 1.3; attribute the author and platform and avoid verbatim reuse beyond short cited references.
Abstract Algebra: Theory and Applications (Judson), Section 6.2: Lagrange's Theorem — LibreTexts Mathematics. Accessed 2026-08-15. Judson's Abstract Algebra on LibreTexts is released under the GNU Free Documentation License 1.3; attribute the author and platform and avoid verbatim reuse beyond short cited references.

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