Which natural number nearest to 8485 is completely divisible by 75?

Introduction / Context:
This question asks for the natural number nearest to 8485 that is divisible by 75. To solve it, we must identify multiples of 75 close to 8485 and compare their distances from this target. This tests understanding of multiples, division with remainders and how to choose the nearest value.

Given Data / Assumptions:

- The reference number is 8485. - We seek a natural number that is a multiple of 75. - We want the multiple closest to 8485 in absolute difference.

Concept / Approach:
To find the nearest multiple of 75, we divide 8485 by 75 and look at the quotient and remainder. The quotient tells us the largest multiple of 75 less than or equal to 8485, and the next integer quotient gives the smallest multiple of 75 greater than 8485. By comparing the difference between 8485 and these two neighboring multiples, we can decide which is closer.

Step-by-Step Solution:
Step 1: Divide 8485 by 75 to find the nearest multiples. Step 2: Compute 75 * 100 = 7500 and 75 * 10 = 750, so 75 * 113 = 75 * (100 + 13) = 7500 + 975 = 8475. Step 3: The next multiple is 75 * 114. Compute 75 * 114 = 75 * (100 + 14) = 7500 + 1050 = 8550. Step 4: The two closest multiples around 8485 are 8475 and 8550. Step 5: Find the distance from 8485 to these multiples. For 8475, difference is 8485 − 8475 = 10. Step 6: For 8550, difference is 8550 − 8485 = 65. Step 7: Since 10 is less than 65, 8475 is closer to 8485 than 8550.

Verification / Alternative check:
We can also check other options quickly. Option B is 8500, and 8500 ÷ 75 = 113 with remainder 25, so it is not a multiple of 75. Option D is 8525, and 8525 ÷ 75 leaves a remainder of 50. Only 8475 and 8550 are exact multiples of 75 among the likely nearby numbers, and as shown, 8475 is closer.

Why Other Options Are Wrong:
Options B, C and D are either not divisible by 75 or not the nearest multiple. Specifically, 8500 and 8525 are not multiples of 75, and although 8550 is a multiple, it is farther away from 8485 than 8475 is.

Common Pitfalls:
Some learners may try to round 8485 to the nearest multiple of 100 or 50 instead of finding exact multiples of 75. Others may not check divisibility properly and choose a number that is close but not a correct multiple. Carefully computing 75 times nearby integers and comparing distances avoids such mistakes.

Final Answer:
The natural number nearest to 8485 that is completely divisible by 75 is 8475, which matches option A.