The Exercise

For positive integers $x$ and $N$ with no common factor, the order of $x$ modulo $N$, written $\operatorname{ord}_N(x)$, is the least positive integer $r$ with $$x^r \equiv 1 \pmod{N}.$$(1) It is the period of the sequence $x, x^2, x^3, \dots$ taken modulo $N$, the quantity that quantum order finding estimates.

The exercise takes $N = 21$ and $x = 5$, which share no factor, and asks to show that the order is $6$. $$5^1 \equiv 5,\quad 5^2 \equiv 4,\quad 5^3 \equiv 20,\quad 5^4 \equiv 16,\quad 5^5 \equiv 17,\quad 5^6 \equiv 1 \pmod{21},$$(2)

The Claim

The library states the claim in the ring $\mathbb{Z}_{21}$ of integers modulo $21$, where the congruence (1) is the equation $5^r = 1$, so that the order of $5$ modulo $21$ is the order of the element $5$ of that ring.

What Is Checked

The claim is stated for the element $5$ of the ring of integers modulo $21$.