The product of my numerator and my denominator is divisible by 2 .my denominator and my numerator differ by 3. My numerator is an even number .my denominator is a prime number.I am:______

If the numerator is even and differs by 3 from the denominator, then the denominator is odd. (Thus, the prime 2 is ruled out.)

At that point, saying that their product is even is redundant.

odd primes are 3,5,7,11,13,...

So, the fraction could be

2/5, 4/7, 14/11, ...

To solve this problem, let's go step by step.

Step 1: The product of the numerator and denominator is divisible by 2.
This means that either the numerator or the denominator, or both, must be divisible by 2. Since the numerator is an even number, it must be divisible by 2.

Step 2: The denominator and numerator differ by 3.
This means that the numerator should be 3 more than the denominator, or the denominator should be 3 less than the numerator. Let's consider the latter case where the numerator is 3 more than the denominator.

Step 3: The numerator is an even number.
Since the numerator is an even number and it is 3 more than the denominator, the denominator must be an odd number.

Step 4: The denominator is a prime number.
Since the denominator is a prime number and it is odd, we can infer that it must be one of the following prime numbers: 3, 5, 7, 11, 13, 17, 19, 23, 29, and so on.

Putting all the information together, the possible solutions for this problem are:
- Numerator: Any even number greater than 3
- Denominator: 3, 5, 7, 11, 13, 17, 19, 23, 29, or any other odd prime number.

In conclusion, based on the given information, you can be any rational number with an even numerator and an odd prime number denominator.

To find the answer, we need to look for a fraction where the numerator and denominator meet the given conditions. Let's break down the given information:

1. The product of the numerator and denominator is divisible by 2.
This means either the numerator or the denominator must be divisible by 2.

2. The numerator and denominator differ by 3.
This means the numerator must be 3 units greater or smaller than the denominator.

3. The numerator is an even number.
This means the numerator must be divisible by 2.

4. The denominator is a prime number.
This means the denominator must be a number greater than 1 that is only divisible by 1 and itself.

Based on these conditions, let's consider different possibilities:

- If we start with an even number as the numerator, such as 2, 4, 6, and so on, we need to find a prime number that is either 3 units greater or smaller than the numerator. For example:
- 2/5: The numerator is even, the denominator is prime, and their difference is not 3.
- 4/7: The numerator is even, the denominator is prime, and their difference is 3. This meets all the conditions.

So, the answer is 4/7.