To solve the problem of finding a pair of integers whose product is -15 and whose difference is 8, we can follow these steps:
### Step 1: Understand the conditions
We need to find two integers, let's call them \( x \) and \( y \), such that:
1. \( x \times y = -15 \) (the product is -15)
2. \( x - y = 8 \) (the difference is 8)
### Step 2: Determine possible pairs for the product
Since the product is negative, one integer must be positive and the other must be negative. We can list the factor pairs of -15:
- \( (1, -15) \)
- \( (-1, 15) \)
- \( (3, -5) \)
- \( (-3, 5) \)
- \( (5, -3) \)
- \( (-5, 3) \)
### Step 3: Check each pair for the difference condition
Now we will check which of these pairs has a difference of 8.
1. **Pair (1, -15)**:
- Difference: \( 1 - (-15) = 1 + 15 = 16 \) (not 8)
2. **Pair (-1, 15)**:
- Difference: \( -1 - 15 = -16 \) (not 8)
3. **Pair (3, -5)**:
- Difference: \( 3 - (-5) = 3 + 5 = 8 \) (this works!)
4. **Pair (-3, 5)**:
- Difference: \( -3 - 5 = -8 \) (not 8)
5. **Pair (5, -3)**:
- Difference: \( 5 - (-3) = 5 + 3 = 8 \) (this works!)
6. **Pair (-5, 3)**:
- Difference: \( -5 - 3 = -8 \) (not 8)
### Step 4: List the valid pairs
From our checks, we found two pairs that satisfy both conditions:
- \( (3, -5) \)
- \( (5, -3) \)
### Conclusion
Thus, the required pairs of integers whose product is -15 and whose difference is 8 are:
- \( 3 \) and \( -5 \)
- \( 5 \) and \( -3 \)
