If n is an even integer, which of the following must be an odd integer?

Answer: 3(n+1) is always odd.

• 3n - 2: Let n = 2k. Then 3n - 2 = 6k - 2 = 2(3k - 1), which is even, so this expression is not odd.

• 3(n+1): Let n = 2k. Then n + 1 = 2k + 1 (odd). 3 × (odd) is odd, so 3(n+1) is always odd.

• n - 2: Let n = 2k. Then n - 2 = 2(k - 1), which is even, so this expression is not odd.

• n/3: This may not be an integer for an even n. If n is divisible by 3, then n = 6k and n/3 = 2k (even). Thus n/3 cannot be guaranteed to be odd.

Therefore the expression 3(n+1) is the one that must be odd for every even integer n.