Understanding Factorial and Last Non-Zero Digit
To find the last non-zero digit of 96!, we must consider how trailing zeros are formed and how to determine the last non-zero digit after removing these zeros.
Trailing Zeros in Factorials
- Trailing zeros in a factorial are created by pairs of 2s and 5s from the prime factorization.
- In 96!, the number of 5s is less than the number of 2s, thus limiting the number of trailing zeros.
- Count the number of factors of 5:
- 96/5 = 19
- 96/25 = 3
- Total = 19 + 3 = 22 trailing zeros.
Removing Trailing Zeros
- To find the last non-zero digit, we can ignore the factors of 10 (which are made up of 2 and 5).
- We need to calculate the product of the digits from 1 to 96, excluding multiples of 10, while keeping track of the surplus 2s.
Calculating Last Non-Zero Digit
- Starting from the non-zero digits, multiply all numbers from 1 to 96, but ignore any factors of 10.
- Count remaining factors of 2 after pairing with 5s:
- Factors of 2 are abundant, so we have more than 22 remaining.
Final Calculation
- After computing the product and reducing by factors of 2 and 5, the last non-zero digit can be determined through modular arithmetic.
- The last non-zero digit of 96! is found to be 6.
Conclusion
- The correct answer for the last non-zero digit of 96! is option 'D' (6).