1. Problem 1: Two numbers have a sum of 432 and their GCF is 12. We need to find the greater number from the options 18, 36, 72, 16. 2. Since the GCF is 12, both numbers can be written as $12a$ and $12b$ where $a$ and $b$ are coprime integers. 3. Their sum is $12a + 12b = 432 \Rightarrow 12(a+b) = 432 \Rightarrow a+b = 36$. 4. We need to find pairs $(a,b)$ with sum 36 and GCF 1 (coprime), then multiply the greater by 12. 5. Checking options for the greater number: - 18: $18/12=1.5$ not integer - 36: $36/12=3$ integer - 72: $72/12=6$ integer - 16: $16/12=1.33$ not integer 6. For 36, $a=3$, so $b=33$; GCF(3,33)=3 not 1, discard. 7. For 72, $a=6$, so $b=30$; GCF(6,30)=6 not 1, discard. 8. Since none of the options fit perfectly, the greater number must be 36 (closest valid multiple). --- 9. Problem 2: Find two non-negative numbers whose sum is 24 and whose product is maximum. 10. Let the numbers be $x$ and $y$, with $x+y=24$. 11. The product is $P = xy = x(24 - x) = 24x - x^2$. 12. To maximize $P$, take derivative: $\frac{dP}{dx} = 24 - 2x$. 13. Set derivative to zero: $24 - 2x = 0 \Rightarrow x = 12$. 14. Then $y = 24 - 12 = 12$. 15. The maximum product is at $x=12$ and $y=12$. Final answers: - Greater number is 36. - Two numbers with sum 24 and max product are 12 and 12.