The nearest integer to 58701 which is exactly divisible by 567 is

### Concept & Logic "Nearest integer" problems require evaluating two candidate multiples: the one immediately below the base number, and the one immediately above it. After dividing the base by the divisor to find the remainder $R$, we compare the distances: * Distance to lower multiple $= R$ * Distance to upper multiple $= \text{Divisor} - R$ The required answer corresponds to whichever distance is numerically smaller. ### Step-by-Step Solution * **Step 1:** Divide the base number by the divisor to find the remainder. $58701 \div 567$ * **Step 2:** Execute long division. $587 \div 567 = 1$. Remainder is $20$. Bring down $0$ to make $200$. $200 \div 567 = 0$. Remainder is $200$. Bring down $1$ to make $2001$. $2001 \div 567 = 3$. ($567 \times 3 = 1701$). Remainder $= 2001 - 1701 = 300$. * **Step 3:** Calculate the two distances. Distance down $= 300$ Distance up $= 567 - 300 = 267$ * **Step 4:** Compare the distances to find the nearest integer. Since $267 < 300$, the upper multiple is strictly closer to the base number. We find it by adding the upward distance: $58701 + 267 = 58968$. ### Exam Strategy & Shortcut **Block Approximation:** Look at the first three digits of the dividend and divisor. We know $567 \times 100 = 56700$. Let's subtract this large chunk from our target. $58701 - 56700 = 2001$. Now we only need to find the nearest multiple of $567$ to $2001$. $567 \times 3 = 1701$ (Gap of $300$) $567 \times 4 = 2268$ (Gap of $267$) The multiple $2268$ is closer to $2001$. So, $56700 + 2268 = 58968$ is the nearest integer. This avoids formal long division entirely. ### Common Pitfall A major trap is always subtracting the remainder by default, which yields the nearest *lower* integer ($58701 - 300 = 58401$). The phrase "nearest integer" is bidirectional. You must always check whether rounding up or rounding down provides a shorter absolute distance. ### Final Answer **Therefore, the correct answer is 58968.**