A teacher multiplies 987 by a certain whole number and obtains 556781 as the product. It is later found that the digits 6 and 7 in this answer are wrong but the other digits are correct. What is the correct product?

Introduction:
This question combines number sense with error detection. The teacher multiplied 987 by an unknown whole number and wrote the product incorrectly, with exactly two wrong digits. Your task is to identify which option could be the correct product under these conditions.

Given Data / Assumptions:

• Supposed product written by the teacher: 556781.
• The digits 6 and 7 in this written answer are wrong.
• The other digits (5, 5, 8, 1) are correct and in the correct positions.
• The true product must be a multiple of 987.

Concept / Approach:
We know:

• Only two digits are incorrect in the written product.
• Any correct answer must match 556781 in all other digit positions.
• The correct product must be divisible by 987.

So we examine all options that differ from 556781 only in the digits 6 and 7 and check which one is divisible by 987.

Step-by-Step Solution:
Step 1: Compare options digit by digit. Original incorrect product: 5 5 6 7 8 1.
We look for an option that keeps positions 1, 2, 5, and 6 (5, 5, 8, 1) the same and changes only the middle digits.

• 553681: 5 5 3 6 8 1 (two middle digits differ, but we must also check divisibility).
• 555181: 5 5 5 1 8 1 (two middle digits differ).
• 556581: 5 5 6 5 8 1 (only one of the wrong digits changed).
• 555681: 5 5 5 6 8 1 (two middle digits changed from 6,7 to 5,6).
• 557681: 5 5 7 6 8 1 (again two middle digits changed).

Several options change exactly two digits, so we must use the divisibility condition next.
Step 2: Check divisibility by 987. We check which candidate is an exact multiple of 987. When tested, 555681 is divisible by 987 with no remainder, while the other options do not divide exactly by 987.
Thus 555681 is the only candidate that both changes exactly the wrong digits and is a valid product of 987 and some whole number.

Verification / Alternative Check:
Compute: 555681 / 987 = 563. So the teacher was really multiplying 987 by 563. Then recompute: 987 * 563 = 555681, which matches option 555681 exactly.

Why Other Options Are Wrong:
Although some options adjust two digits, they fail the divisibility test by 987 and therefore cannot be correct products of 987 and an integer. Others may implicitly change digits that were supposed to be correct. Only 555681 both respects which digits are wrong and satisfies the multiplication condition.

Common Pitfalls:
Students sometimes ignore the requirement that the result must be a multiple of 987 or misinterpret which digits are incorrect. Always enforce both conditions: the digit pattern and the exact divisibility by the known multiplier.

Final Answer:
The correct product of 987 and the whole number is 555681.