Explanation:

- Even Power of Odd Integers Greater than 1: When you raise an odd integer (greater than 1) to an even power, the result will also be odd. Examples include numbers like 3^2 = 9, 5^4 = 625, etc.

- Division by 8: Odd numbers raised to even powers, when divided by 8, often yield a remainder of 1. This is due to properties like (2k+1)^2 = 1 (mod 8), as squaring odd numbers commonly results in numbers equating to 1 mod 8.

- Explanation of Options:

- Option 1 (Remainder is 3): Incorrect, as squaring or raising to any even power yields 1, not 3.

- Option 2 (Remainder is 2): Incorrect for similar reasons; the remainder heads towards 1 not 2.

- Option 3 (Remainder is 1): This is typically the correct behavior, as explained in modular arithmetic.

- Option 4 (Remainder is not necessarily 1): This option is misleading; the remainder is usually 1.

- Option 3 is Correct: The remainder when any even power of an odd number is divided by 8 is indeed typically 1.