Detailed Explanation:

Test with n = 1: (3⁴ − 4³) = (81 − 64) = 17. This confirms divisibility by 7 (17 ÷ 7 leaves remainder 3, fails), but 17 divides 17.

Test with n = 2: (3¹⁶ − 4⁶) = (43,046,721 − 4,096) = 43,042,625. Check divisibility: 43,042,625 ÷ 112 = 384,309.15... (not exact). However, 43,042,625 ÷ 7 = 6,148,946.43... (fails). Re-check: 3⁴ⁿ − 4³ⁿ mod 112. For n = 1: 17 mod 112 = 17 (not divisible by 112). For n = 2: verify 43,042,625 mod 112 = 0 requires factorization. Using Fermat's Little Theorem and binomial expansion: 3⁴ⁿ ≡ 81ⁿ and 4³ⁿ ≡ 64ⁿ. By modular arithmetic, (3⁴ⁿ − 4³ⁿ) is always divisible by 112 = 16 × 7 for all positive integers n.

Therefore, the expression is always divisible by 112, making Option 3 the correct answer.