In the product 75? × 49 × 867 × 943, the units digit of the result is 1. Determine the missing digit ? in 75?.

Introduction / Context:
Determining a missing units digit often relies on modular arithmetic base 10. Only the units digits of the factors affect the final units digit of a product, which allows a compact calculation without full multiplication.

Given Data / Assumptions:

• Number forms: 75? has units digit d (unknown).
• Units digits of the other factors: 49 → 9, 867 → 7, 943 → 3.
• The final product’s units digit equals 1.

Concept / Approach:
Compute the product of units digits modulo 10. Multiply progressively and keep only the last digit. Then solve for d such that d * (last-digit-product) ≡ 1 (mod 10). This uses the idea of multiplicative inverses modulo 10 for the final stage.

Step-by-Step Solution:

Verification / Alternative check:
Replace ? with 9: units(759 * 49 * 867 * 943) = units(9 * 9 * 7 * 3) = units(9 * 9) = 1 after full modular multiplication, confirming the requirement.

Why Other Options Are Wrong:
Digits 1, 3, 5, or 7 fail d * 9 ≡ 1 (mod 10); their products have units 9, 7, 5, and 3 respectively, not 1.

Common Pitfalls:
Multiplying entire numbers unnecessarily or forgetting that only units digits matter for determining the final units digit.

Final Answer:
9