Determinant and Divisibility

The rows of a determinant of a \(3\times 3\) matrix consist of three consecutive digits of certain three-digit numbers, all of which are divisible by \(17\). Prove that the determinant is also divisible by \(17\).

Solution

Let the determinant \(D\) be

\[ D= \left | \begin{array}{ccc} a_{11}&a_{12}&a_{13} \\ a_{21}&a_{22}&a_{23} \\ a_{31}&a_{32}&a_{33} \end{array} \right | . \]

Applying column operations we get successively

\[ 10^{3}D = \left | \begin{array}{ccc} 10^{2}a_{11}&10a_{12}&a_{13} \\ 10^{2}a_{21}&10a_{22}&a_{23} \\ 10^{2}a_{31}&10a_{32}&a_{33} \end{array} \right | = \left | \begin{array}{ccc} 10^{2}a_{11}+10a_{12}+a_{13} & 10a_{12}&a_{13} \\ 10^{2}a_{21}+10a_{22}+a_{23} & 10a_{22}&a_{23} \\ 10^{2}a_{31}+10a_{32}+a_{33} & 10a_{32}&a_{33} \end{array} \right | . \]