Answer to Question #12455 in Linear Algebra for john.george.milnor

Question

Show that determinant
2 0 6 0 4
5 3 2 2 7
2 5 7 5 5
2 0 9 2 7
0 0 2 8 9
is divisible by 17

Answer 1

The determinant will have no changes after adding one column to another,
multiplied by some number.
So we can
1)multiply first column by 10000 and
add to the last one
2)multiply second column by 1000 and add to the last
one
3)multiply third column by 100 and add to the last one
4)multiply
forth column by 10 and add to the last one

After this elementary column
operations last column will be

20604
53227
25755
20927
289
All these numbers are divisible
by 17, so as one column is divisible by 17,
then the whole determinant is
divisible by 17 too.