1. **Problem Statement:** Joanne's calculator has a broken 8 button. She wants to calculate $82 \times 816$ using this calculator. 2. **Method to solve with broken 8 button:** - Since the 8 button is broken, Joanne cannot directly input numbers containing 8. - However, she can use the distributive property of multiplication to avoid pressing 8. - Express $82$ as $80 + 2$ and $816$ as $800 + 16$. 3. **Using distributive property:** $$82 \times 816 = (80 + 2) \times (800 + 16)$$ $$= 80 \times 800 + 80 \times 16 + 2 \times 800 + 2 \times 16$$ 4. **Calculate each term separately:** - $80 \times 800 = 64000$ (no 8 button needed if 80 and 800 can be input as $8 \times 10$ and $8 \times 100$, but since 8 is broken, input 80 as $10 \times 8$ is not possible. Instead, use $82 = 80 + 2$ is problematic because 80 contains 8. So, better to use a different approach. 5. **Alternative approach:** - Use multiplication by 82 as $82 = 100 - 18$. - Calculate $82 \times 816 = (100 - 18) \times 816 = 100 \times 816 - 18 \times 816$ - Joanne can input 100 and 816 (no 8 in 100), but 816 contains 8, so she cannot input 816 directly. 6. **Better approach:** - Express 816 as $800 + 16$. - $82 \times 816 = 82 \times (800 + 16) = 82 \times 800 + 82 \times 16$ - 82 contains 8, so cannot input 82 directly. 7. **Use multiplication by 2 and 80 separately:** - $82 = 80 + 2$, but 80 contains 8. 8. **Use multiplication by 2 and 80 as $8 \times 10$ but 8 is broken. So, input 80 as $10 \times 8$ is not possible. 9. **Use multiplication by 2 and 80 as $4 \times 20$ but 4 is not broken in part (a), so 4 can be used. 10. **Calculate $82 \times 816$ as:** - $82 = 4 \times 20 + 2$ (since 4 and 2 buttons work) - So, $82 \times 816 = (4 \times 20 + 2) \times 816 = 4 \times 20 \times 816 + 2 \times 816$ 11. **Calculate $4 \times 20 \times 816$:** - $4 \times 20 = 80$ (8 button broken, but 4 and 2 work, so input 4 and 20 separately) - Then multiply 80 by 816 (80 contains 8, so again problem) 12. **Use repeated addition or multiplication by smaller numbers:** - Since 8 button is broken, Joanne can multiply by 2 and 4, which work. - For example, multiply 82 by 816 as $82 \times 816 = 82 \times (800 + 16)$ - Input 800 as $8 \times 100$ not possible, but 800 can be input as $100 \times 8$ no 8 button. 13. **Conclusion for part (a):** - Joanne can use the fact that $82 \times 816 = (80 + 2) \times (800 + 16)$ and calculate each term separately by multiplying numbers without 8. - She can multiply 2 by 16 and 2 by 800 (input 2 and 16, 2 and 800), and multiply 80 by 16 and 80 by 800 by using multiplication by 10 and 8 replaced by multiplication by 4 and 2 (since 4 and 2 buttons work). 14. **Part (b):** If both 8 and 4 buttons are broken, Joanne cannot use the above method because 4 is needed to break down 8. - Without 8 and 4 buttons, she cannot input numbers containing 8 or 4. - Since 82 and 816 both contain 8, and 4 is needed to break down 8, she cannot calculate $82 \times 816$ using the calculator. **Final answer:** - (a) Use distributive property and break down 8 into 4 and 2 to multiply without pressing 8. - (b) No, if both 8 and 4 buttons are broken, Joanne cannot calculate the product using the calculator.