Units digit reasoning — Determine the units digit in the product (784 * 618 * 917 * 463). Focus only on the units-place multiplication rules.

Introduction / Context:
Finding the units digit of a large product is a classic speed-math task. You do not need the full product; you only need the units digits of the factors and modular arithmetic mod 10.

Given Data / Assumptions:

• Numbers: 784, 618, 917, 463.
• We only use each number’s units digit.
• We work modulo 10 for the units place.

Concept / Approach:
The units digit of a product depends only on the units digits of the factors. Compute stepwise using modular arithmetic: (a * b) mod 10, then multiply by the next units digit, continuing until done.

Step-by-Step Solution:
Extract units digits: 784 → 4, 618 → 8, 917 → 7, 463 → 3.Compute 4 * 8 = 32 → units 2.Now 2 * 7 = 14 → units 4.Now 4 * 3 = 12 → units 2.Final units digit = 2.

Verification / Alternative check:
Use modular arithmetic: ((((4*8) mod 10)*7) mod 10)*3 mod 10 = 2, giving the same result efficiently.

Why Other Options Are Wrong:

• 3 / 4 / 5 / 6: Each arises from an intermediate step or guess but not from the complete chain of mod 10 multiplications.

Common Pitfalls:
Multiplying entire numbers (time-consuming); stopping after an intermediate units digit (e.g., after two factors) instead of finishing all four factors; arithmetic slips when reducing mod 10.

Final Answer:
2