Two integers are chosen at random and multiplied. Find the probability that the product is an even integer.

Explanation

To solve the problem of finding the probability that the product of two randomly chosen integers is an even integer, we can follow these steps:

### Step 1: Understand the Types of Integers
There are two types of integers: even and odd.
- Even integers are those that are divisible by 2 (e.g., 0, 2, 4, 6, ...).
- Odd integers are those that are not divisible by 2 (e.g., 1, 3, 5, 7, ...).

### Step 2: Identify Possible Cases
When we multiply two integers, there are three possible cases regarding their parity (evenness or oddness):
1. Both integers are even.
2. Both integers are odd.
3. One integer is even and the other is odd.

### Step 3: Analyze Each Case
- **Case 1:** If both integers are even, their product is even (e.g., 2 × 4 = 8).
- **Case 2:** If both integers are odd, their product is odd (e.g., 3 × 5 = 15).
- **Case 3:** If one integer is even and the other is odd, their product is even (e.g., 2 × 3 = 6).

### Step 4: Count the Total Outcomes
From the above cases, we can summarize:
- Total cases = 3 (as identified in the three cases).

### Step 5: Count the Favorable Outcomes
The favorable outcomes for the product being even are:
- Case 1 (both even) and Case 3 (one even, one odd).
Thus, there are 2 favorable outcomes (Case 1 and Case 3).

### Step 6: Calculate the Probability
The probability \( P \) that the product of the two integers is even is given by the formula:
\[ P(\text{even product}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2}{3} \]

### Conclusion
Therefore, the probability that the product of two randomly chosen integers is an even integer is \( \frac{2}{3} \).

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