How to Find the First and Last Digits of a Large Factorial Power

When dealing with extremely large numbers, such as 1443!^{2022}, we often need to understand certain properties of these numbers. This includes finding both the first and last non-zero digits. This article explores the methods and computations involved in finding these digits, primarily using logarithms for simplification.

Introduction to Finding the First Non-Zero Digit

We have previously determined that the first non-zero digit of 1443!^{2022} is 5. The process involves several steps using logarithms and specific formulas. For a number n, we express it as n f cdot 10^k, where f in [1, 10). The first digit of n is the integer part of 10^{log_{10}n - lfloor log_{10}n rfloor}.

Formula and Calculation

The formula for finding the first digit is as follows:

s_1 sum_{k2}^{1443} lfloor log_{10}k rfloor s_2 sum_{k2}^{1443} (log_{10}k - lfloor log_{10}k rfloor) n s_1 lfloor s_2 rfloor r_1 s_2 - lfloor s_2 rfloor m 2022n lfloor 2022r_1 rfloor r_2 2022r_1 - lfloor 2022r_1 rfloor x 10^{r_2}

Using these steps, we find:

1443!^{2022} approx x 10^m

This gives us the result: 5.519603871 times 10^{7954776}, which is accurate for the left-most 10 digits.

Identifying the Right-Most Non-Zero Digits

The integer part of the number ends with 0149451776, followed by 723876 zeros. Therefore, the full number can be represented as:

5519603871...014945177600000...00000

Key Points: The last 723876 zeros align with the power of 10 component. The sequence 0149451776 precedes these zeros and represents the last non-zero digits.

General Method for Finding the First Digit

For any number n geq 1, we can express it in the form n f cdot 10^k, where f in [1, 10). This allows us to find the first digit by evaluating 10^{log_{10}n - lfloor log_{10}n rfloor}.

Example Calculation for a General Number

Consider a number n and express it as:

n 1443!^{2022}

Using the logarithm property, we compute:

log_{10}1443!^{2022} 2022 sum_{j1}^{1443} log_{10}j 7954776.74

This shows that the number has 7954776 1 7954777 digits, and the first digit is:

lfloor 10^{0.74} rfloor 5

This can be verified using a calculator or a program with sufficient precision.

Conclusion

The process of finding the first and last non-zero digits of extremely large numbers like 1443!^{2022} involves logarithmic calculations and series summations. By breaking down the problem, we can handle the computational complexity and extract meaningful properties from these numbers.