Evaluating the divisibility of a sum of large exponential terms

Factor the bases using the repdigit constant 111 = 3 × 37: 222 = 2 × 111 = 2 × 3 × 37 333 = 3 × 111 = 3 × 3 × 37

Test for 2: 222^333 is an even number, but 333^222 is an odd number raised to a power, which remains completely odd. An even number plus an odd number always results in an odd sum, meaning the entire expression cannot be divisible by 2.

Test for 3 and 37: Both bases are explicit multiples of 111 (3 × 37). Because every base contains these prime factors, both individual exponential components are divisible by 3 and 37, making their combined sum divisible by 3 and 37 as well.

Answer: (b).