Final answer:
The statement is true. An even integer squared remains even and multiplied by three stays even. Conversely, if the product of the square of a number and three is even, the original number must also be even.
Step-by-step explanation:
The statement that a number n is an even integer if and only if 3n² is an even integer is True. Let's consider n to be an even integer. By definition, an even integer is any integer that can be divided by 2 without leaving a remainder, so n can be expressed as 2k where k is an integer. When we square n to obtain n², we get (2k)² which is equal to 4k². Multiplying this by 3 gives us 12k², which is also divisible by 2, indicating that 3n² is an even number. Conversely, if 3n² is even, then n² is also even since an odd number times 3 would still yield an odd number and not an even number. Therefore, n must be even as squares of odd integers are odd.