Understanding the Remainder of a Sum of Factorials Divided by 8

In this article, we delve into the process of finding the remainder when the sum of the factorial series from 1! to 1000! is divided by 8. This involves a clear understanding of how factorials grow and their properties when taken modulo 8. Understanding this concept can help in various fields, from number theory to computer science.

Factorial Basics and Modulo Operations

A factorial of a number n, denoted as n!, is the product of all positive integers up to n. For instance, 4! 4 * 3 * 2 * 1 24.

Properties of Factorials Modulo 8

Let's first analyze the factorials from 1 to 1000 modulo 8:

1! 1 2! 2 3! 6 4! 24 ≡ 0 (mod 8) 5! 120 ≡ 0 (mod 8) 6! 720 ≡ 0 (mod 8) 7! 5040 ≡ 0 (mod 8) 8! 40320 ≡ 0 (mod 8) For n ≥ 4, n! will always be divisible by 8 since n! contains at least three factors of 2 (2, 4, and another even number).

Sum of Factorials Modulo 8

Given the rapid growth of factorials, it's only necessary to check the sum of the first three factorials when calculating the sum modulo 8:

s ≡ 1! 2! 3! 4! 5! 6! 7! 8! ... 1000! (mod 8)

Since all factorials from 4! to 1000! are divisible by 8, we can simplify this as:

s ≡ 1! 2! 3! 0 0 0 0 ... 0 (mod 8)

Calculating the relevant terms:

s ≡ 1 2 6 9 (mod 8)

Finally, finding the remainder when 9 is divided by 8:

9 ≡ 1 (mod 8)

Thus, the remainder when the sum s is divided by 8 is 1.

Alternative Approach: Simplifying the Calculation

Given that 4! 4 * 3 * 2 * 1 is divisible by 8, and all larger factorials must be divisible by 8, it simplifies our calculation to the sum of the first three factorials. Therefore:

s 1! * 2! * 3! (mod 8)

Substituting the values:

s 1 * 2 * 3 6 (mod 8)

However, the sum of 1! 2! 3! 1 2 6 9, and when taken modulo 8, gives:

9 ≡ 1 (mod 8)

Conclusion

From the above analysis, we can see that the remainder when the sum of factorials from 1! to 1000! is divided by 8 is 1. This result is consistent with computational verification using the J programming language:

8 / !1 to 100 1

This method uses the properties of factorials and modulo operations to simplify a complex problem. It's a great example of how understanding number properties can lead to efficient problem solving. Whether you're a student or a professional in mathematics, computer science, or any field reliant on number theory, this concept can provide valuable insights.

Keywords: factorial, modulo, remainder