To solve \(2^{23} \mod 53\), we can again use the method of successive squaring.

We've already calculated some powers of 2 modulo 53 in the previous problem. Let's build on that.

1. \(2^2 \mod 53 = 4\)
2. \(2^4 \mod 53 = 16\)
3. \(2^8 \mod 53 = 44\)
4. \(2^{16} \mod 53 = 28\)

Now, let's find \(2^{23} \mod 53\):

1. First, calculate \(2^{23}\) as:
   \[
   2^{23} = 2^{16} \times 2^4 \times 2^2 \times 2^1
   \]

2. Using the known results:
   \[
   2^{23} \mod 53 = (2^{16} \mod 53) \times (2^4 \mod 53) \times (2^2 \mod 53) \times (2 \mod 53)
   \]
   \[
   = 28 \times 16 \times 4 \times 2 \mod 53
   \]

3. Compute step by step:
   \[
   28 \times 16 = 448 \quad \Rightarrow \quad 448 \mod 53 = 448 - 8 \times 53 = 448 - 424 = 24
   \]
   \[
   24 \times 4 = 96 \quad \Rightarrow \quad 96 \mod 53 = 96 - 53 = 43
   \]
   \[
   43 \times 2 = 86 \quad \Rightarrow \quad 86 \mod 53 = 86 - 53 = 33
   \]

So, \(2^{23} \mod 53 = 33\).

Would you like more details or have any questions?

Here are some related questions:
1. What is \(2^{25} \mod 53\)?
2. How can you use modular exponentiation in encryption algorithms like RSA?
3. How would you calculate \(3^{23} \mod 53\)?
4. What is the role of modular arithmetic in solving congruences?
5. Can you explain the importance of the order of elements in modular arithmetic?
6. What is \(2^{100} \mod 53\)?
7. How do you find the modular inverse of a number modulo a prime?
8. How would \(2^{23} \mod 53\) change if 53 were not prime?

**Tip:** Remember that in modular arithmetic, intermediate results can be reduced modulo the base at each step to simplify calculations and prevent working with large numbers.

Learn how to calculate 2 to the power of 23 modulo 53 using modular arithmetic and successive squaring. This problem involves Fermat's Little Theorem and is suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation