If a₁b₁c₁, a₂b₂c₂, and a₃b₃c₃ are three-digit even natural numbers and $\mathrm{\Delta }=\left|\begin{array}{l}{\mathrm{c}}_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{a}}_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{b}}_1\\ {\mathrm{c}}_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{a}}_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{b}}_2\\ {\mathrm{c}}_3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{a}}_3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{b}}_3\end{array}\right|$, then $∆$ is

As a₁b₁c₁, a₂b₂c₂, and a₃b₃c₃ are even natural numbers, each of c₁, c₂, c₃ is divisible by 2. Let ${\mathrm{c}}_{\mathrm{i}}=2{\mathrm{k}}_{\mathrm{i}}$ for i = 1, 2, 3. Thus,

$\mathrm{\Delta }=2\left|\begin{array}{l}{\mathrm{k}}_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{a}}_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{b}}_1\\ {\mathrm{k}}_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{a}}_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{b}}_2\\ {\mathrm{k}}_3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{a}}_3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{b}}_3\end{array}\right|=2\mathrm{m}$

where m is some natural number. Thus, $∆$ is divisible by 2.That $∆$ may not be divisible by 4 can be seen by taking the three numbers as 112, 122, and 134. Note that

$\mathrm{\Delta }=\left|\begin{array}{l}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\\ 4\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\end{array}\right|=2$

which is divisible by 2 but not bY 4.