For n a nonnegative integer either n 0 mod 3 or n 1 mod 3 or

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 (that is, the lowest possible non-negative integer) modulo 3 of the expressions given: Since n^3 + 2n 0 mod 3 for all n, we conclude that 3 divides n^3 + 2n for any nonnegative integer n. 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.

Solution

The correct answer is Option A for below part, since (n^3+2n) will be divisible by 3 for all n

n (mod 3)	n^3 mod(3)	2n (mod 3)	(n^3+2n) mod 3
0	0	0	0
1	1	2	0
2	2	1	0