A teacher multiplies 987 by a certain integer and gets 556781 as the product. It is known that in this answer only the digits 6 and 7 are wrong, and all the other digits are correct. What is the correct product?

Introduction / Context:
This problem combines multiplication with error detection. We know that the teacher tried to multiply 987 by some integer and obtained a six digit result, but exactly two digits in the middle of the product are wrong. Our task is to identify the correct product using the information that the positions of the incorrect digits are fixed and the rest of the digits are correct. This type of question tests logical reasoning and understanding of divisibility.

Given Data / Assumptions:

• The intended multiplication is 987 times some integer k.
• The incorrect reported product is 556781.
• Only the digits 6 and 7 in this six digit result are wrong. The digits 5, 5, 8, and 1 in their positions are correct.
• Therefore, the correct product must have the form 55XY81, where X and Y are digits replacing 6 and 7.
• The correct product must be exactly divisible by 987.

Concept / Approach:
We know the correct answer must match the pattern 55XY81 and must be a multiple of 987. Instead of guessing the unknown integer k directly, we can test the given answer options, all of which fit the general pattern of keeping the first two and last two digits similar. We simply divide each option by 987 and see which one gives an integer with no remainder. The one that divides exactly will be the correct product. Because the options are close together, this approach is efficient.

Step-by-Step Solution:
Step 1: Note that the wrong answer is 556781. Only the third and fourth digits (6 and 7) may change.Step 2: The correct answer must be of the form 55XY81, matching the first two and last two digits.Step 3: Examine each option: 553681, 555181, 556581, and 555681 (plus any extra distractor) to see which is divisible by 987.Step 4: Divide 553681 by 987; it does not give an integer, so it is not correct.Step 5: Divide 555181 by 987; again it does not divide exactly.Step 6: Divide 556581 by 987; still not a whole number.Step 7: Divide 555681 by 987; this division gives an exact integer quotient (555681 ÷ 987 = 563), with remainder zero.Step 8: Therefore the correct product must be 555681.

Verification / Alternative check:
To cross check, compute 987 × 563 explicitly. First 987 × 500 = 493500, then 987 × 60 = 59220, and 987 × 3 = 2961. Add these: 493500 + 59220 = 552720, then 552720 + 2961 = 555681. This matches the candidate option exactly, confirming that 555681 is the true product when 987 is multiplied by 563. Since this product has the digits 5, 5, 5, 6, 8, 1 and only the third and fourth digits differ from 556781, the error description is satisfied.

Why Other Options Are Wrong:
Option 553681: Even though it looks similar to the wrong answer, it is not divisible by 987 exactly, so it cannot be the correct product.Option 555181: This also fails to be a multiple of 987; division leaves a nonzero remainder.Option 556581: Dividing by 987 again results in a non integer; it does not correspond to 987 multiplied by an integer k.Option 556351: This option neither fits the specific error pattern nor divides exactly by 987.

Common Pitfalls:
Students may misread the statement and assume that any two digits may be wrong, not specifically the digits 6 and 7 in the middle. Others might try to guess the integer multiplier without using divisibility, leading to unnecessary effort. Some may also forget to test divisibility completely and rely only on visual similarity. Always use the key condition that the correct answer must be a multiple of 987 and must differ only in the specified digit positions.

Final Answer:
The correct product of 987 and the unknown integer is 555681.