b. Represent the original number as n , and use deductive reasoning to prove the conjecture in part (a). Multiply the number by 8 . Add 12 to the product.

To prove the conjecture using deductive reasoning, let's follow the steps outlined in the problem. 1. **Define the original number**: Let the original number be represented as \( n \). 2. **Multiply the number by 8**: \[ 8n \] 3. **Add 12 to the product**: \[ 8n + 12 \] Now, we can analyze the expression \( 8n + 12 \). ### Step-by-step reasoning: - **Step 1**: We start with the original number \( n \). - **Step 2**: When we multiply \( n \) by 8, we get \( 8n \). - **Step 3**: Adding 12 to this product gives us \( 8n + 12 \). ### Conclusion: The expression \( 8n + 12 \) can be factored to show a relationship: \[ 8n + 12 = 8n + 8 \cdot 1.5 = 8(n + 1.5) \] This shows that the result \( 8n + 12 \) is always a multiple of 8, regardless of the value of \( n \). Thus, we can conclude that the conjecture holds true: the result of multiplying the original number \( n \) by 8 and then adding 12 will always yield a number that is a multiple of 8.