Explanation:

- Divisibility by 2:

- Any odd number raised to any power remains odd.

- Since both 555 and 777 are odd, 555⁷⁷⁷ and 777⁵⁵⁵ are also odd.

- Sum of two odd numbers is even.

- So, 555⁷⁷⁷ + 777⁵⁵⁵ is divisible by 2.

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- Divisibility by 3:

- If the sum of digits of a number is divisible by 3, the number is divisible by 3.

- Sum of digits of 555 = 5 + 5 + 5 = 15, divisible by 3.

- Sum of digits of 777 = 7 + 7 + 7 = 21, divisible by 3.

- Both 555⁷⁷⁷ and 777⁵⁵⁵ are divisible by 3.

- Sum is divisible by 3.

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- Divisibility by 37:

- 555 is divisible by 37 (555 = 37 × 15).

- Thus, 555⁷⁷⁷ is divisible by 37.

- 777 is also divisible by 37 (777 = 37 × 21).

- Thus, 777⁵⁵⁵ is divisible by 37.

- Sum is divisible by 37.

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- Therefore, the expression is divisible by 2, 3, and 37.

- Correct option: 4, 1, 2, and 3.

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