Question

If $x_1=a_i{b}_i{c}_i,i=1,2,3$ are three-digit positive integers such that each $x_1$ is a multiple of $19$, then for some integer $n$, prove that $\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|$ is divisible by $19$.

Answer 1

we can write a three digit number as $abc=a\times {10}^2+b\times 10+c$

$\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|$

just multiply ${10}^2$ in the first row and $10$ in the second row and add 3 rows which results the first row into

$\left|\begin{array}{ccc}{a}_1{b}_1{c}_1& a_2{b}_2{c}_2& a_3{b}_3{c}_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|$

which makes the first row three elements as multiples of 19.

$\therefore$ hence proved.