Number System — Find the units digit (last digit) of the product 41 × 42 × 43 × 44 × 45 × 46 × 47 × 48 × 49. Clearly indicate only the final units digit required; no intermediate expansion is necessary.

Introduction / Context:
This problem tests quick reasoning about units digits (the last digit) in large products. Rather than multiplying all numbers from 41 through 49, we rely on modular thinking and the presence of key factors that force the units digit to become 0.

Given Data / Assumptions:

• We need only the units digit of 41 × 42 × 43 × 44 × 45 × 46 × 47 × 48 × 49.
• Basic property: any integer multiple of 10 ends with units digit 0.
• A multiple of 10 occurs when the product contains at least one factor 2 and one factor 5.

Concept / Approach:
To get a units digit of 0, it is enough that the product include factors 2 and 5 simultaneously. Among consecutive integers, even numbers supply many 2s, and any number ending in 5 supplies a factor 5. Therefore we can decide the last digit without full multiplication.

Step-by-Step Solution:
1) Scan the list 41..49 for a factor 5: 45 ends with 5, so it contributes a factor 5.2) Scan the list for at least one even number (factor 2): 42, 44, 46, 48 are even, providing multiple 2s.3) Since the product contains both 2 and 5, it contains the factor 10 = 2 * 5.4) Any product divisible by 10 must end with the digit 0. Hence the units digit is 0.

Verification / Alternative check:
Even if 45 were replaced by another number ending with 5 within the range, the conclusion would remain the same because there are multiple even numbers to pair with it. One 2 and one 5 guarantee a trailing zero.

Why Other Options Are Wrong:
1, 4, 5, and 6 cannot be the last digit once the product is a multiple of 10. A units digit other than 0 would imply the product is not divisible by 10, contradicting the presence of 2 and 5.

Common Pitfalls:
Attempting full multiplication wastes time; forgetting that one 2 and one 5 suffice; overlooking that several even numbers are present; confusing tens digit with units digit.

Final Answer:
0