`" if " x_(i) =a_(i) b_(i) C_(i), i= 1,2,3` are three- digit positive integer such that each `x_(i)` is a mulptiple of 19 then prove that det`{{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(c_(1),,c_(2),,c_(3)):}}` is divisible by 19.

Explanation

`Delta = |{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),((100a_(1)+10b_(1)+c_(1)),,(100a_(2)+10b_(2)+c_(2)),,(100a_(3)+10b_(3)+c)(3)):}|`
`|{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(x_(1),,x_(2),,x_(3)):}|= |{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(19m_(1),,19m_(2),,19m_(3)):}|`
`19=|{:(a_(1) ,,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(m_(1),,m_(2),,m_(3)):}|=19n`
where `n= |{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(m_(1),,m_(2),,m_(3)):}|` is certainly an integer.