A boy multiplied 987 by a number and got 559981. Both 9s in the product are wrong; the other digits are correct. What is the correct product?

Given data

• Incorrect product shown as 559981; digits except the two 9s are correct ⇒ true product has the form 55xy81.
• We need 987 × N = 55xy81 for some integer N.

Concept / Approach

• Use modular arithmetic on ending digits to constrain N.

Step-by-step calculation

Last digit condition: 987 ≡ 7 (mod 10). To end with 1, 7N ≡ 1 (mod 10) ⇒ N ≡ 3 (mod 10).Last two digits 81: 987 ≡ −13 (mod 100). Solve −13N ≡ 81 (mod 100) ⇒ 13N ≡ 19 (mod 100).Inverse of 13 mod 100 is 77 (13×77 = 1001 ≡ 1). Thus N ≡ 19×77 ≡ 1463 ≡ 63 (mod 100).Try N = 563 (closest three-digit ending with 63 near the ballpark).Compute: 987 × 563 = 987×500 + 987×60 + 987×3 = 493500 + 59220 + 2961 = 555681.

Verification

Pattern matches 55xy81 with x=5, y=6; the two 9s in 559981 were the incorrect digits.

Common pitfalls

• Assuming only one pair of 9s could be wrong; using modular constraints quickly nails the correct product.

Final Answer

555681