Use indirect reasoning to show that there is no whole number, \( n \), which solves the equation \( 6 n+14=31 \). Is the number 31 even or odd? odd even Is the number 14 even or odd? even odd Is the number \( 6 n \) even or odd? even Is the sum \( 6 n+14 \) even or odd? even odd

To show that there is no whole number \( n \) that solves the equation \( 6n + 14 = 31 \) using indirect reasoning, we can analyze the parity (evenness or oddness) of the numbers involved. 1. **Identify the parity of the numbers:** - The number \( 31 \) is **odd**. - The number \( 14 \) is **even**. - The term \( 6n \) is **even** because \( 6 \) is even and the product of an even number and any whole number is even. 2. **Analyze the left side of the equation:** - The left side of the equation is \( 6n + 14 \). - Since \( 6n \) is even and \( 14 \) is even, the sum \( 6n + 14 \) is also **even** (the sum of two even numbers is even). 3. **Compare the parities:** - The left side \( 6n + 14 \) is even. - The right side \( 31 \) is odd. 4. **Conclusion:** - An even number cannot equal an odd number. Therefore, there is no whole number \( n \) that can satisfy the equation \( 6n + 14 = 31 \). Now, let's summarize the answers to the questions: - Is the number \( 31 \) even or odd? **Odd** - Is the number \( 14 \) even or odd? **Even** - Is the number \( 6n \) even or odd? **Even** - Is the sum \( 6n + 14 \) even or odd? **Even**