Solving Modular Arithmetic: Remainder of a Complex Exponential Series When Divided by 8

Modular arithmetic is a fundamental concept in number theory, often appearing in real-world applications such as cryptography, computer science, and competitive programming. This article delves into a specific problem involving exponential series and modular arithmetic, demonstrating step-by-step how to approach such problems.

Problem Statement

Given the series:

S 31 #8290; 3123 #8290; 31233 #8290; #8230; #8290; 3123#8290;#8230;#8290;1002

the task is to find the remainder when this series is divided by 8.

Preliminary Steps

First, we rewrite the series in a more manageable form:

S 31 #8290; 32212 #8290; 33312 #8290; #8230; #8290; 310010012

where each term is (3^{frac{n(n-1)}{2}}).

Step 1: Simplify (3^k mod 8)

By calculating various powers of 3 modulo 8, we observe the following pattern:

3^1 #8764; 3 mod 8 3^2 #8764; 9 mod 8 1 3^3 #8764; 3 cdot 3^2 mod 8 3 cdot 1 mod 8 3 3^4 #8764; 3^2 cdot 3^2 mod 8 1 mod 8

The pattern shows that:

(3^{2k-1} equiv 3 mod 8) (3^{2k} equiv 1 mod 8)

Step 2: Determine the Parity of (frac{n(n-1)}{2})

We need to determine if (frac{n(n-1)}{2}) is odd or even.

If (n 2k), then (n-1 2k-1), so (frac{n(n-1)}{2}) is even. Hence, (3^{frac{n(n-1)}{2}} equiv 1 mod 8). If (n 2k-1), then (n-1 2k-2), so (frac{n(n-1)}{2}) is odd. Hence, (3^{frac{n(n-1)}{2}} equiv 3 mod 8).

Step 3: Counting Even and Odd n

From 1 to 100, there are 50 odd numbers and 50 even numbers. This means:

50 terms contribute 3 each to the series. 50 terms contribute 1 each to the series.

So, the contributions to the series (S) are:

Total from even (n): (50 times 1 50) Total from odd (n): (50 times 3 150)

Step 4: Calculate the Total Contribution to (S)

The total sum (S) is:

150 50 200

Now, we find (200 mod 8):

200 div 8 25 Exactly 25 200 equiv 0 mod 8

Thus, the remainder when (S) is divided by 8 is 0, confirming that 200 is divisible by 8.

Conclusion

The remainder when the given series is divided by 8 is: 0.

Additional Verification

Using the J programming language, the verification can be done as:

8 3x^1 3 63x^/:i.100 0

This confirms that the solution is correct and follows the pattern we have established.

Key Takeaways

Understanding the pattern in powers of 3 modulo 8. Determining the parity of the exponent (frac{n(n-1)}{2}). Counting the contributions of even and odd exponents.

Keywords

modular arithmetic, exponentiation, remainder theorem