Use modular arithmetic to reduce the exponent

Oct 13, 2025 by talynt.org 137 views

Introduction

In this article, we will delve into the world of elementary number theory and modular arithmetic to prove that $16^{16}$ ends with $1616$ . This problem may seem daunting at first, but with the right approach, we can break it down into manageable steps and arrive at a solution without resorting to 'extraneous' calculations.

The Problem Statement

We are given the expression $\frac{16^{16} - 1616}{10000}$ and asked to prove that it is an integer. In other words, we need to show that $16^{16}$ ends with $1616$ . To approach this problem, we will use modular arithmetic and the properties of congruences.

Modular Arithmetic Basics

Before we dive into the solution, let's briefly review some basics of modular arithmetic. Given an integer $a$ and a positive integer $n$ , we say that $a$ is congruent to $b$ modulo $n$ if $a \equiv b \pmod{n}$ . This is denoted as $a \equiv b \pmod{n}$ . We can also write $a \equiv b \pmod{n}$ as $a - b$ is divisible by $n$ .

The Solution

To prove that $16^{16}$ ends with $1616$ , we will use the following approach:

Express $16^{16}$ in terms of $2^{64}$ : We can write $16^{16}$ as $(2^4)^{16} = 2^{64}$ .
Use modular arithmetic to reduce the exponent: We will use the properties of congruences to reduce the exponent $64$ modulo $10000$ .
Show that the reduced exponent is equivalent to $4$ modulo $10000$ : We will use the fact that $2^4 \equiv 16 \pmod{10000}$ to show that the reduced exponent is equivalent to $4$ modulo $10000$ .
Conclude that $16^{16}$ ends with $1616$ : We will use the fact that $2^4 \equiv 16 \pmod{10000}$ to conclude that $16^{16}$ ends with $1616$ .

Step 1: Express $16^{16}$ in terms of $2^{64}$

We can write $16^{16}$ as $(2^4)^{16} = 2^{64}$ .

Step 2: Use modular arithmetic to reduce the exponent

We will use the properties of congruences to reduce the exponent $64$ modulo $10000$ . We can write $64 \equiv 4 \pmod{10000}$ .

Step 3: Show that the reduced exponent is equivalent to $4$ modulo $10000$

We will use the fact that $2^4 \equiv 16 \pmod{10000}$ to show that the reduced exponent is equivalent to $4$ modulo $10000$ . We can write:

2^{64} \equiv (2^4)^{16} \equiv 16^{16} \pmod{10000}

Step 4: Conclude that $16^{16}$ ends with $1616$

We will use the fact that $2^4 \equiv 16 \pmod{10000}$ to conclude that $16^{16}$ ends with $1616$ . We can write:

16^{16} \equiv 2^{64} \equiv (2^4)^{16} \equiv 16^{16} \pmod{10000}

This shows that $16^{16}$ ends with $1616$ .

Conclusion

In this article, we have proven that $16^{16}$ ends with $1616$ using modular arithmetic and the properties of congruences. We have shown that the exponent $64$ can be reduced modulo $10000$ to $4$ , and that the resulting expression is equivalent to $16^{16}$ modulo $10000$ . This proves that $16^{16}$ ends with $1616$ .

References

[1] Elementary Number Theory by David Burton
[2] Modular Arithmetic by Keith Devlin

Additional Information

This problem was inspired by a discussion on the math stack exchange website. The original problem statement was:

"Prove that $16^{16}$ ends with $1616$ without calculating $2^{64}$ ."

Q&A: Frequently Asked Questions

Q: What is the main goal of this article?

A: The main goal of this article is to prove that $16^{16}$ ends with $1616$ using modular arithmetic and the properties of congruences.

Q: Why is this problem important?

A: This problem is important because it demonstrates the power of modular arithmetic in solving complex problems. By using modular arithmetic, we can reduce the exponent $64$ modulo $10000$ and show that $16^{16}$ ends with $1616$ .

Q: What is modular arithmetic?

A: Modular arithmetic is a system of arithmetic that "wraps around" after reaching a certain value, called the modulus. In this case, we are working modulo $10000$ , which means that any number greater than $10000$ is equivalent to its remainder when divided by $10000$ .

Q: How does modular arithmetic help us solve this problem?

A: Modular arithmetic helps us solve this problem by allowing us to reduce the exponent $64$ modulo $10000$ . This reduces the problem to a simpler one, where we can use the properties of congruences to show that $16^{16}$ ends with $1616$ .

Q: What is the significance of the number $10000$ in this problem?

A: The number $10000$ is significant in this problem because it is the modulus we are working with. Any number greater than $10000$ is equivalent to its remainder when divided by $10000$ .

Q: Can we use this method to prove that any number $a^n$ ends with $a^n$ ?

A: No, we cannot use this method to prove that any number $a^n$ ends with $a^n$ . This method relies on the specific properties of the number $16$ and the modulus $10000$ . In general, it is not possible to prove that any number $a^n$ ends with $a^n$ using this method.

Q: What are some potential applications of this method?

A: This method has potential applications in cryptography, coding theory, and other areas of mathematics where modular arithmetic is used. It can also be used to solve other problems involving large exponents and modular arithmetic.

Q: Can you provide more examples of how to use this method?

A: Yes, here are a few more examples of how to use this method:

Prove that $2^{32}$ ends with $16$ .
Prove that $3^{64}$ ends with $81$ .
Prove that $4^{128}$ ends with $256$ .

Conclusion

In this article, we have provided a Q&A section to answer some of the most frequently asked questions about the problem of proving that $16^{16}$ ends with $1616$ . We have also provided some additional information and examples of how to use this method.

References

[1] Elementary Number Theory by David Burton
[2] Modular Arithmetic by Keith Devlin

Additional Information

This problem was inspired by a discussion on the math stack exchange website. The original problem statement was:

"Prove that $16^{16}$ ends with $1616$ without calculating $2^{64}$ ."

The solution presented in this article uses modular arithmetic and the properties of congruences to prove that $16^{16}$ ends with $1616$ .