Explanation:

Let's analyze 222³³³ + 333²²² for divisibility:

- Divisibility by 2: Any odd number raised to any power is odd, and any even number raised to a positive power is even. Here, 222³³³ is even, 333²²² is odd. So, even + odd = odd. Thus, it's not divisible by 2.

- Divisibility by 3:

- 222 = 0 mod 3, so 222³³³ = 0 mod 3.

- 333 = 0 mod 3, so 333²²² = 0 mod 3.

- So their sum is divisible by 3.

- Divisibility by 37:

- Note: 222 = -1 mod 37, 333 = 1 mod 37.

- 222³³³ = (-1)³³³ = -1 mod 37.

- 333²²² = 1²²² = 1 mod 37.

- So sum = -1 + 1 = 0 mod 37. Thus, divisible by 37.

Options:

1. 2 and 3 but not 37 –

2. 3 and 37 but not 2 –

3. 2 and 37 but not 3 –

4. 2, 3 and 37 –

Correct Answer:

Option: 2, 3 and 37 but not 2