Question about determinants.

Suppose you have n numbers of the form (a1+n*a2+n^2*a3+...n^(n-1)*an) where a1, a2, a3, ..., an are all integers in the interval [0,n-1]. Suppose that d divides all of those numbers. Suppose you arrange those a1, a2, a3, ..., an in each row of a matrix. Does d necessarily divide the determinant of that matrix?

You mean evenly divisible?  In that case yes, it would imply that exactly, because the determinant is the sum of the products, all of which are divisible (evenly?) by d.  This comes down to a problem of:

[(a1*b1) + (a2*b2)]/d = (a1*b1)/d + (a2*b2)/d.

If the original numbers are factors of d, then their products are factors of d, and the sum of those products must also be a factor of d. 

Proof is more than likely in any linear mathematics text.  Should be fairly straight forward.