A boy multiplied 987 by a certain integer but wrote the product as 559981. If only the digits 9 and 8 in his written answer are incorrect and all other digits are correct, what is the correct product?

Introduction / Context:
This question tests error analysis in multiplication. We are told that a boy multiplied 987 by some integer but recorded the product incorrectly as 559981, with the additional information that only certain digits in the written product are wrong and the rest are correct. Our task is to reconstruct the correct product using this limited information and logical checking.

Given Data / Assumptions:
Multiplier: 987.Incorrect recorded product: 559981.We are told that some digits are wrong while others are correct.We must choose the correct product from the given options.

Concept / Approach:
One straightforward way is to test each option by dividing it by 987. The correct product must be an exact multiple of 987, giving an integer quotient with no remainder. Because 987 is large, mental calculation is hard, but basic long division or approximate checks can be used. Among the close numerical options, only the exact multiple of 987 can represent the true product.

Step-by-Step Solution:
Step 1: Consider the answer choices as possible correct products: 553681, 555181, 555681 and 556581.Step 2: Divide 553681 by 987. The division leaves a remainder, so this is not a correct product.Step 3: Divide 555181 by 987. This also leaves a remainder, so it is not a correct product.Step 4: Divide 556581 by 987. Again, the result is not an integer, so this option is incorrect.Step 5: Divide 555681 by 987. This division gives exactly 563 with no remainder.Step 6: Therefore, the correct product of 987 and the integer 563 is 555681.Step 7: The incorrect product 559981 presumably arose from miswriting some of the digits of 555681, specifically the positions where 9 and 8 appear.

Verification / Alternative check:
We can perform the multiplication 987 * 563 directly to confirm. Multiplying 987 by 563 using standard long multiplication yields 555681. Since this exactly matches one of the options, and that option is the only exact multiple of 987 among the choices, it confirms that 555681 is the correct product.

Why Other Options Are Wrong:
Each of the other options, when divided by 987, leaves a nonzero remainder. That means they cannot be the true product of 987 and an integer. Even if their digits look similar to the incorrect product 559981, they do not satisfy the fundamental multiplicative requirement. Only 555681 passes the exact divisibility test.

Common Pitfalls:
A common mistake is to try to adjust the digits of 559981 by eye without checking actual divisibility, which can lead to picking an option that looks similar but is not an exact multiple of 987. Another error is to assume that the difference between the incorrect and correct products must be small and ignore the requirement that the result must be a perfect product.

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