Let \(\alpha\) be a nonzero algebraic integer and let \(n\) be a positive integer. Define \(d_n(\alpha)\) as the maximal positive integer \(d\) for which \(\alpha^n \equiv 1 \pmod{d}\). The author proves that \(d_{\gcd(m,n)}(\alpha)= \gcd(d_m(\alpha),d_n(\alpha))\) for all \(m,n \in {\mathbb N}\). By definition, this means that \(d_n(\alpha),\) \(n=1,2,3,\dots,\) is a strong divisibility sequence. (For instance, the Fibonacci sequence is also a strong divisibility sequence.) He also studies the growth of this divisibility sequence \(d_n(\alpha)\) and proves that \(\lim_{n\to \infty} {\log d_n(\alpha)\over n}=0,\) unless some power of \(\alpha\) is a rational integer or a unit in a quadratic extension of \({\mathbb Q}\). With the same restrictions on \(\alpha,\) he conjectures that \(d_n(\alpha)=d_1(\alpha)\) for infinitely many positive integers \(n\) and gives some numerical evidence to support this conjecture for \(\alpha\) being a root of \(x^3-x-1.\) The case when \(\alpha\) is a quadratic unit is studied in detail. In particular, for the unit \(\alpha=u+v \sqrt{D}\) associated to a nontrivial positive solution of the Pell equation \(u^2-v^2D=1\), the number \(d_n(\alpha)\) is computed explicitly. It turns out that then \(\lim_{n\to \infty} {\log d_n(\alpha)\over n}={1\over 2} \log \alpha,\) so \(d_n(\alpha)\) grows rapidly.