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Modular Arithmetic: Solving 7^126 Modulo 48
Modular Arithmetic: Solving 7^126 Modulo 48
Modular arithmetic is a fundamental concept in mathematics with numerous applications in computer science, cryptography, and number theory. One common problem involves finding the remainder when a large number is divided by another. This article explores the process of determining what the remainder is when (7^{126}) is divided by 48. Specifically, we will demonstrate that the remainder is 1 using modular exponentiation and properties of congruences.
Introduction to Modular Arithmetic
Modular arithmetic deals with the concept of congruence, where two integers are considered equivalent if they have the same remainder when divided by a given modulus. The modulus here is 48. We use the notation (a equiv b pmod{m}) to express that (a) and (b) are congruent modulo (m).
Understanding the Problem
Given the problem: What is the remainder when (7^{126}) is divided by 48?
We can express this mathematically as:
[7^{126} pmod{48}]The goal is to simplify this expression to find the remainder efficiently.
Using Modular Properties
A key property of modular arithmetic is the congruence of exponents. If we can find that (7^2 equiv 1 pmod{48}), then we can simplify the problem significantly. Let's verify this step by step.
Step 1: Calculate (7^2 mod 48)
We start by calculating the square of 7 modulo 48:
[7^2 49]When we divide 49 by 48, the remainder is 1:
[49 equiv 1 pmod{48}]This confirms that:
[7^2 equiv 1 pmod{48}]Step 2: Applying the Property of Exponents
Since (7^2 equiv 1 pmod{48}), we can use this to simplify higher powers of 7. Notice that:
[7^{126} (7^2)^{63}]Using our congruence, we have:
[(7^2)^{63} equiv 1^{63} pmod{48}]Since any power of 1 is still 1, we get:
[1^{63} equiv 1 pmod{48}]Therefore:
[7^{126} equiv 1 pmod{48}]Conclusion
The remainder when (7^{126}) is divided by 48 is:
[boxed{1}]Application of Modular Arithmetic
This method of simplification through congruence properties is not only useful for this specific problem but is a powerful tool in many areas of mathematics. It can be applied in cryptography for tasks such as encrypting and decrypting messages, as well as in algorithms for computer science.