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.

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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.
Let the original number be $$ n $$. Multiply it by 8 to get $$ 8n $$, then add 12 to obtain $$ 8n + 12 $$. This expression shows that the result is always a multiple of 8, proving the conjecture.

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.

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