a. Write a conjecture that relates the result of the process to the original number selected. Represent the original number as \( n \). The result is 4 n . (Simplify your answer.) b. Represent the original number as n , and use deductive reasoning to prove the conjecture in part (a). Multiply the number by 8 .

If you multiply an original number \( n \) by 8, the conjecture states that the result is equal to \( 4n \) if you halve the product. In simplified terms, this means that: Original operation: \( 8n \) Halving the result: \( \frac{8n}{2} = 4n \) Hence, the relationship can be expressed as \( \frac{8n}{2} = 4n \), confirming that indeed the outcome is half of eight times the original number. To prove this using deductive reasoning: Start with the original number \( n \). When you multiply it by 8, you get \( 8n \). By applying the division of 2 to this product, it yields \( \frac{8n}{2} \). By simplifying \( \frac{8n}{2} \), you arrive at \( 4n \), thereby confirming the conjecture that the result of halving the product of the original number \( n \) multiplied by 8 is indeed \( 4n \). Thus, the conjecture holds true through basic arithmetic operations.