Question

Calculate $2^{125}$ mod 127 using any method you choose. (Hint: 127 is prime.)

Name: Problem 25, Chapter 1: Algorithms
Uploaded: 2020-06-01T16:35:07-08:00
Duration: 45 s
Channel: Katherine Mcgovern
Description: Calculate 2^125 mod 127 using any method you choose. (Hint: 127 is prime.)

   Calculate $2^{125}$ mod 127 using any method you choose. (Hint: 127 is prime.)

Algorithms

Sanjoy Dasgupta,… 1st Edition

Chapter 1, Problem 25

Instant Answer

Solved by Expert Katherine Mcgovern

06/01/2020

Step 1

In other words, a^(p-1) ≡ 1 (mod p). In our case, p = 127, so we have 2^(127-1) ≡ 1 (mod 127). Now, we want to find 2^125 (mod 127). We can rewrite 2^125 as 2^(126-1). Using Fermat's Little Theorem, we know that 2^126 ≡ 2 (mod 127). Show more…

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Calculate $2^{125}$ mod 127 using any method you choose. (Hint: 127 is prime.)

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Key Concepts

Modular Arithmetic

Modular arithmetic is the branch of arithmetic dealing with integers and the operations of addition, subtraction, and multiplication with respect to a fixed modulus. It is used to simplify calculations by focusing on the remainders when numbers are divided by the modulus.

Fermat’s Little Theorem

Fermat’s Little Theorem states that if p is a prime number and a is an integer not divisible by p, then a^(p-1) is congruent to 1 modulo p. This theorem provides a powerful tool for reducing large exponents modulo a prime, thereby simplifying computations like 2^125 mod 127.

Modular Exponentiation

Modular exponentiation is a method used for efficiently computing large exponents in a modular system, often by techniques such as exponentiation by squaring. It leverages the properties of modular arithmetic and theorems like Fermat’s Little Theorem to reduce the size of the numbers involved, making large power computations manageable.

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