Question 103
For all integers $x$, $y$, $z$, set
$$
S(x, y, z) = (xy - xz, yz - yx, zx - zy).
$$
Given any integers $a$, $b$, $c$ with $abc \gt 1$, show that there exist integers $n_0$ and $k$ with $0 \lt k \le abc$ such that
$$
S^{n+k}(a, b, c) \equiv S^n(a, b, c) \pmod{abc}
$$
for all integers $n \ge n_0$.
$(S^1 = S$ and, for each integer $m \ge 1, S^{m+1} = S \circ S^m$. $(u_1, u_2, u_3) \equiv (v_1, v_2, v_3)(\text{mod} M) \Leftrightarrow u_i \equiv v_i(\text{mod} M)(i = 1, 2, 3).)$
$$
S(x, y, z) = (xy - xz, yz - yx, zx - zy).
$$
Given any integers $a$, $b$, $c$ with $abc \gt 1$, show that there exist integers $n_0$ and $k$ with $0 \lt k \le abc$ such that
$$
S^{n+k}(a, b, c) \equiv S^n(a, b, c) \pmod{abc}
$$
for all integers $n \ge n_0$.
$(S^1 = S$ and, for each integer $m \ge 1, S^{m+1} = S \circ S^m$. $(u_1, u_2, u_3) \equiv (v_1, v_2, v_3)(\text{mod} M) \Leftrightarrow u_i \equiv v_i(\text{mod} M)(i = 1, 2, 3).)$