Math (Algebra)
posted by Drake .
Find the largest integer N for which N−6 evenly divides N^3−6.

N^3−6 is zero Modulo (N6). If we compute Modulo (N6) then obviously:
N6 = 0 >
N = 6
Here and in the following the equals sign means equality modulo N  6.
We then have:
N^3 6 = 6^3  6 = 210
Therefore:
210 = 0
Reverting back to the ordinary definition of the equals sign, this means that:
210 = k (N6)
So, N6 must be the largest possible factor of 210, which is 210 therefore
N = 216.