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Consider the fact that $25875, 46552, 41354, 48691, 95818$ are all divisible by $23$. Use this fact to determine if \begin{vmatrix} 2 &5 &8 &7 &5 \\ 4 &6 &5 &5 &2 \\ 4 &1 &3 &5 &4 \\ 4 &8 &6 &9 &1 \\ 9 &5 &8 &1 &8 \end{vmatrix} is divisible by 23 or not, without directly evaluating the determinant.

Pak Long
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    You do not change the value of a determinant when you add to a column a linear combination of other columns. Can you exploit this fact by adding a well-chosen combination to the right column? – TerranDrop Dec 09 '15 at 10:00
  • I think it is $10000 C_{1} + 1000 C_{2} + 100 C_{3} + 10 C_{4} + C_{5}$, so the right column is \begin{matrix} 25875 \ 46552 \ 41354 \ 48691 \ 95818 \end{matrix} – Pak Long Dec 09 '15 at 10:05

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A determinant of a matrix can be looked at as an (antisymmetric) multilinear function on the rows of that matrix. So:

$\det(23r_1,r_2,r_3,r_4,r_5)=23\det(r_1,r_2,r_3,r_4.r_5)$

drhab
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