Let $Z_n$ be the group of integers $\{0, 1, 2, \dots, n - 1\}$ with addition modulo $n$ as the group operation. The number of elements in the group $Z_2 \times Z_3 \times Z_4$ that are their own inverses is _________

An element $(x, y, z) \in Z_2 \times Z_3 \times Z_4$ is its own inverse (self-inverse) if and only if:

This splits into three independent congruences:

1. In $(Z_2, +_2)$:

   $$2x \equiv 0 \pmod 2$$

   This is true for all $x \in Z_2 = \{0, 1\}$. Thus, there are 2 solutions ($x = 0, 1$).

2. In $(Z_3, +_3)$:

   $$2y \equiv 0 \pmod 3$$

   Since $\gcd(2, 3) = 1$, the only solution is $y = 0$. Thus, there is 1 solution.

3. In $(Z_4, +_4)$:

   $$2z \equiv 0 \pmod 4$$

   This holds for $z = 0$ and $z = 2$. Thus, there are 2 solutions.

By the product rule, the total number of self-inverse elements in $Z_2 \times Z_3 \times Z_4$ is:

The elements are $(0, 0, 0)$, $(1, 0, 0)$, $(0, 0, 2)$, and $(1, 0, 2)$.