The number nearest to 43582 divisible by each of 25, 50 and 75 is

### Concept & Formula To find the number nearest to a target value $T$ that is divisible by a group of numbers, compute the Least Common Multiple (LCM) of those numbers. Next, divide $T$ by the LCM to find the remainder, allowing you to determine both the closest lower and upper bounding multiples. $$\text{Lower Multiple} = T - \text{Remainder}$$ $$\text{Upper Multiple} = \text{Lower Multiple} + \text{LCM}$$ Compare both to see which absolute distance is smaller. ### Step-by-Step Solution **Given:** * Target number = $43582$ * Divisors: $25, 50, 75$ **Calculation:** * Step 1: Find the LCM of $25, 50,$ and $75$. * The multiples of $75$ are $75, 150, 225...$ * $150$ is perfectly divisible by both $25$ and $50$. * $\text{LCM}(25, 50, 75) = 150$ * Step 2: Divide $43582$ by $150$ to locate the remainder position. $$43582 \div 150 = 290 \text{ with a remainder of } 82$$ * Step 3: Find the lower bounding multiple. $$\text{Lower Multiple} = 43582 - 82 = 43500$$ * Step 4: Find the upper bounding multiple. $$\text{Upper Multiple} = 43500 + 150 = 43650$$ * Step 5: Evaluate which option is closest to $43582$. * Distance to $43500$: $43582 - 43500 = 82$ * Distance to $43650$: $43650 - 43582 = 68$ * Distance to option (c) $43600$: Notice that $43600$ isn't a standard option in our upper/lower bounds calculated directly via the floor, let's double-check if $43600$ is a multiple of $150$: $43600 \div 150 = 290.66$ (Not a multiple!). Wait, let's verify option values: $43500 = 150 \times 290$. Let's re-evaluate the actual options provided: $43500$ and $43650$ are the actual multiples. Let's check distance to all valid options. * Let's evaluate all options for divisibility by $150$: * (a) $43500 \div 150 = 290$ (Valid) * (b) $43550 \div 150 = 290.33$ (Invalid) * (c) $43600 \div 150 = 290.66$ (Invalid) * (d) $43650 \div 150 = 291$ (Valid) * Comparing valid options (a) and (d) for proximity to $43582$: * $|43582 - 43500| = 82$ * $|43582 - 43650| = 68$ * Since $68 < 82$, $43650$ is the nearest correct number. ### Exam Strategy & Shortcut Filter out invalid options first using the divisibility rule for $150$ (must end in $00$ or $50$ and be divisible by $3$). * Sum of digits test for multiples of $3$: * (a) $4+3+5+0+0 = 12$ (Divisible by $3$) * (b) $4+3+5+5+0 = 17$ (Eliminate) * (c) $4+3+6+0+0 = 13$ (Eliminate) * (d) $4+3+6+5+0 = 18$ (Divisible by $3$) Now choose between $43500$ and $43650$ by finding the closest spatial distance to $43582$. Since $43582$ is visually closer to $43650$ than to $43500$, pick (d) immediately. ### Common Pitfall Students often compute the lower remainder bound ($43500$) and pick it blindly because it is the first choice listed, neglecting to verify if the upper threshold alternative is closer. ### Final Answer **Therefore, the correct answer is 43650.**