answer:To find the divisor, we can use the formula for division: Dividend = (Divisor × Quotient) + Remainder Given: Remainder = 19 Quotient = 61 Dividend = 507 Let's call the Divisor "D". Plugging in the values we have: 507 = (D × 61) + 19 Now, let's solve for D: 507 - 19 = D × 61 488 = D × 61 Now, divide both sides by 61 to find the value of D: D = 488 / 61 D = 8 So, the Divisor is boxed{8} .

answer:For the function y=x^{3}-2ax+a, taking the derivative yields y'=3x^{2}-2a. Since the function y=x^{3}-2ax+a has a minimum value in the interval (0,1), we have y'=3x^{2}-2a=0, which has a root in (0,1). When a > 0, the roots of 3x^{2}-2a=0 are ± sqrt { frac {2}{3}a}. If there is a root in (0,1), then 0 < sqrt { frac {2}{3}a} < 1, which implies 0 < a < frac {3}{2}. When a=0, the roots of 3x^{2}-3a=0 are equal and are both 0, and the function f(x) has no minimum value in (0,1). When a < 0, the equation 3x^{2}-3a=0 has no roots, and the function f(x) has no minimum value in (0,1). Combining all cases, we have 0 < a < frac {3}{2}. Therefore, the answer is boxed{D}. By analyzing the derivative of the function y=x^{3}-2ax+a in the interval (0,1), we find that the derivative has at least one real root in this interval. We then discuss three cases: a > 0, a=0, and a < 0 to determine the range of values for the real number a.

answer:To solve this problem, analyze the given statements to find the one true statement while ensuring the other three are false. Step 1: Assume Statement I is true. - If Emma is older than David, then Statement I is true. - This means Statements II, III, and IV must all be false. - If Statement II (Fiona is not the youngest) is false, Fiona is the youngest. - If Statement III (George is the oldest) is false, George is not the oldest. - If Statement IV (David is not the oldest) is false, David is the oldest. - This configuration results in a contradiction because David cannot be both older and younger than Emma. Step 2: Assume Statement II is true. - If Fiona is not the youngest, then Statement II is true. - This means Statements I, III, and IV must all be false. - If Statement I (Emma is older than David) is false, David is older than or the same age as Emma. - If Statement III (George is the oldest) is false, George is not the oldest. - If Statement IV (David is not the oldest) is false, David is the oldest. - This configuration allows for David being the oldest, Fiona not the youngest, and George not the oldest; Emma is younger than David. Fiona can fit between Emma and George. Step 3: Assume Statement III is true. - If George is the oldest, then Statement III is true. - This means Statements I, II, and IV must all be false. - If Statement I (Emma is older than David) is false, David is older than or the same age as Emma. - If Statement II (Fiona is not the youngest) is false, Fiona is the youngest. - If Statement IV (David is not the oldest) is false, David is the oldest. - This configuration results in a contradiction because George and David cannot both be the oldest. Step 4: Assume Statement IV is true. - If David is not the oldest, then Statement IV is true. - This means Statements I, II, and III must all be false. - If Statement I (Emma is older than David) is false, David is older than or the same age as Emma. - If Statement II (Fiona is not the youngest) is false, Fiona is the youngest. - If Statement III (George is the oldest) is false, George is not the oldest. - This configuration implies that Emma, being older than David but David not the oldest, and with Fiona the youngest and George not the oldest, places Emma as the oldest. This analysis leads to the conclusion that the correct configuration with all consistency is when Statement II is true: David is the oldest, followed by Emma or George, and Fiona is not the youngest. The ranking from oldest to youngest would be David, Emma, George, Fiona. Conclusion: The correct ranking from oldest to youngest is David, Emma, George, Fiona, thus the answer is textbf{(B) text{David, Emma, George, Fiona}}. The final answer is boxed{textbf{(B)} text{David, Emma, George, Fiona}}

answer:Insert values a = 3 and b = 4 into the new operation definition: 3 X 4 = 2(4) + 5(3) - 3^2 - 4 Simplify each part: - 2(4) = 8 - 5(3) = 15 - 3^2 = 9 - Subtract 4 Calculate: 8 + 15 - 9 - 4 = 23 - 9 = 10 Thus, boxed{10} Conclusion: The arithmetic is correct and the steps follow the defined operation, which makes the new problem solvable and consistent with the new operation definition.