For n a nonnegative integer, either n ≡ 0 mod 3 or n ≡ 1 mod 3 or n ≡ 2 mod 3. In each case, fill out the following table with the canonical representatives modulo 3 of the expressions given:
n mod 3 | n^3 mod 3 | 2n mod 3 | n^3 + 2n mod 3
0 | | |
1 | | |
2 | | |
From this, we can conclude:
A. Since n^3 + 2n ≡ 0 mod 3 for all n, we conclude that 3 divides n^3 + 2n for any nonnegative integer n.
B. Since n^3 + 2n ≢ 0 mod 3 for all n, we conclude that 3 does not necessarily divide n^3 + 2n for all nonnegative integers n.