Correct Answer: D) 8 ### Step-by-Step Solution #### Step 1: Formulate the Linear Diophantine Equation Let the number of units of Component $A$ produced be $x$. Let the number of units of Component $B$ produced be $y$. Since $x$ and $y$ represent physical quantities of manufactured components, they must be non-negative integers ($x, y \in \mathbb{Z}_{\ge 0}$). Convert the total allocated assembly time from hours to minutes: $$\text{Total Time} = 11 \text{ hours} \times 60 \text{ minutes/hour} = 660 \text{ minutes}$$ Using the processing time constraints per component, we establish the linear equation: $$18x + 25y = 660 \quad \text{--- (Equation 1)}$$ This is a linear Diophantine equation of the form $ax + by = c$, where $a = 18$, $b = 25$, and $c = 660$. #### Step 2: Analyze the Existence of Integer Solutions A linear Diophantine equation has integer solutions if and only if the Greatest Common Divisor $\text{GCD}(a, b)$ divides $c$. * $\text{GCD}(18, 25) = 1$ (since 18 and 25 share no common prime factors and are co-prime). * Since $1$ perfectly divides $660$, integer solutions are guaranteed to exist. #### Step 3: Find an Initial Base Solution $(x_0, y_0)$ We reduce Equation 1 using modular arithmetic to find a base solution. Taking the equation modulo $18$ (the smaller coefficient) isolates $y$: $$18x + 25y \equiv 660 \pmod{18}$$ Simplifying the coefficients and constants modulo $18$: * $18x \equiv 0 \pmod{18}$ * $25 \equiv 7 \pmod{18}$ * $660 = (18 \times 36) + 12 \implies 660 \equiv 12 \pmod{18}$ Substituting these remainders back into the congruence relation: $$7y \equiv 12 \pmod{18}$$ To solve for $y$, add multiples of $18$ to the right-hand side until the value becomes perfectly divisible by $7$: $$7y \equiv 12 + 18 \equiv 30 \pmod{18}$$ $$7y \equiv 30 + 18 \equiv 48 \pmod{18}$$ $$7y \equiv 48 + 18 \equiv 66 \pmod{18}$$ $$7y \equiv 66 + 18 \equiv 84 \pmod{18}$$ Now, divide both sides by $7$: $$y \equiv 12 \pmod{18}$$ Thus, our initial base integer value for $y$ is $y_0 = 12$. Substitute $y = 12$ back into Equation 1 to find the corresponding value for $x_0$: $$18x + 25(12) = 660$$ $$18x + 300 = 660$$ $$18x = 360 \implies x_0 = 20$$ Our baseline integer coordinate solution is $(x_0, y_0) = (20, 12)$. #### Step 4: Express the General Parameter Solution Space All valid integer solutions for this system progress dynamically along a fixed parametric grid derived from the coefficients: $$x = 20 + 25t$$ $$y = 12 - 18t$$ where $t$ is any integer ($t \in \mathbb{Z}$). #### Step 5: Apply Non-negative Boundary Conditions To maintain physical real-world boundaries, both component quantities must remain non-negative ($x \ge 0$ and $y \ge 0$). * **Boundary condition for $x$:** $$20 + 25t \ge 0 \implies 25t \ge -20 \implies t \ge -0.8$$ * **Boundary condition for $y$:** $$12 - 18t \ge 0 \implies 18t \le 12 \implies t \le 0.67$$ Since $t$ must be an integer falling within the interval $[-0.8, 0.67]$, the only permissible value for the parameter is: $$t = 0$$ Let us check if there are nearby alternative solution shifts. If we invert the signs of the parametric adjustments: $$x = 20 - 25t$$ $$y = 12 + 18t$$ Evaluating bounds for this variation: * **For $x \ge 0$:** $20 - 25t \ge 0 \implies 25t \le 20 \implies t \le 0.8$ * **For $y \ge 0$:** $12 + 18t \ge 0 \implies 18t \ge -12 \implies t \ge -0.67$ This yields $t = 0$ as the unique integer mapping. Thus, there is only one uniquely viable base pair that satisfies the system under basic non-negative constraints. However, let us evaluate the optimization requirement: **minimizing the total number of components produced ($x + y$)**. Let's examine the total production function $S$: $$S = x + y = (20 - 25t) + (12 + 18t) = 32 - 7t$$ To minimize the sum $S$, $t$ must be maximized. The maximum possible integer value for $t$ that satisfies the non-negative boundaries across all structural shifts can be analyzed. Let's systematically list all valid non-negative integer solution pairs for $18x + 25y = 660$: | Solution Case | Component $A$ ($x$) | Component $B$ ($y$) | Total Production ($x + y$) | | --- | --- | --- | --- | | Case 1 | $x = 20$ | $y = 12$ | $20 + 12 = 32$ | Are there other valid non-negative integer pairs? Let's check by changing the base parameter steps: * If we decrease $x$ by 25, it goes negative ($20 - 25 = -5$). * If we increase $x$ by 25, $x = 45$. Then $18(45) + 25y = 660 \implies 810 + 25y = 660 \implies 25y = -150 \implies y = -6$, which violates $y \ge 0$. Hence, $(x, y) = (20, 12)$ is the only valid solution pair in non-negative integers. It automatically represents the minimal operational solution. #### Step 6: Calculate the Absolute Difference The problem asks for the absolute difference between the number of units of Component $A$ and Component $B$ produced: $$\text{Absolute Difference} = |x - y|$$ $$\text{Absolute Difference} = |20 - 12| = 8$$ *Correction and optimization check:* Let us re-verify the full problem boundary parameters for alternative variables. If the minimization condition shifts across different standard boundary zones, the base question text has uniquely isolated $(20, 12)$ as the mandatory anchor layout. Let's test if the original parameter option matching configuration reflects a variation in the objective function layout or parameters. Re-evaluating the question data points: $18x + 25y = 660$. If $x = 5, y = 22.8$ (non-integer). If $x = 20, y = 12$, then absolute difference is $20 - 12 = 8$. Let's analyze if an alternative integer coordinate pair exists if the total allocated capacity is altered. Under the strict parameters provided ($18x + 25y = 660$), the unique solution is $x = 20$ and $y = 12$, yielding a final absolute difference of 8.