Understanding the Unit Digit of n in 1!^4 × 2!^4 × 3!^4 ... × 100!^4

Today, we delve into an intriguing mathematical problem: determining the unit digit of the product of the fourth powers of factorials from 1! to 100!.

Factorials and Their Last Digits

When examining the factorials of integers from 1 to 100, there is a notable trend in their unit digits. Starting from 5! onwards, the unit digits of factorials are consistently 0. This occurs because these factorials include both a 2 and a 5 as factors, resulting in a trailing zero.

Unit Digits of Lower Factorials

Let's break down the unit digits of the lower factorials from 1! to 4!:

1! has a unit digit of 1. 2! has a unit digit of 2. 3! has a unit digit of 6. 4! has a unit digit of 4.

When these factorials are raised to the fourth power, their unit digits are:

(1!) ^4 has a unit digit of 1. (2!) ^4 has a unit digit of 6. (3!) ^4 has a unit digit of 6. (4!) ^4 has a unit digit of 6.

Consistent Unit Digit of Higher Factorials

Since factorials from 5! and beyond end in 0, raising them to any power will also result in a number ending in 0. Therefore, these higher factorials do not change the unit digit of the overall product.

Calculating the Unit Digit of the Product

To find the unit digit of the product of 1!^4, 2!^4, 3!^4, ..., 100!^4, we only need to consider the unit digits of the lower factorials raised to the fourth power. The factorials from 5! to 100! contribute a unit digit of 0, which does not affect the final unit digit of the product.

The unit digits of 1!^4 and 2!^4 contribute to the unit digit of the final product:

1!^4 1^4 1 (unit digit 1) 2!^4 2^4 16 (unit digit 6)

To find the unit digit of the combined product, we calculate:

1 × 6 × a series of 0's, which simplifies to 6, because multiplying any number by 0 results in 0, and 1 × 6 will give us 6 as the unit digit.

However, we must also consider the last factorial, 100!, which influences our result. 100! ends in a large number of 0s (at least 24 zeros due to the factors of 5 and 2). Therefore, (100!^4) will end in even more zeros, (96) to be precise. When we include 100!^4 in our product, it further reinforces the unit digit of 0 for the final result.

Given the extensive zeros from 5!^4 to 100!^4, the unit digit of the entire product is 0. Thus, the unit digit of n (1!^4 × 2!^4 × 3!^4 ... × 100!^4) is 0.

Conclusion

Through a detailed analysis of the unit digits of factorials and their powers, we have established that the unit digit of the product 1!^4 × 2!^4 × 3!^4 ... × 100!^4 is 0. This conclusion is based on the consistent unit digit behavior of factorials beyond 4! and the zero unit digits from 5!^4 onwards.