To solve \(2^{17} \mod 53\), we can use the method of successive squaring:

1. Calculate \(2^2\) mod 53:
   \[
   2^2 = 4
   \]
   \[
   4 \mod 53 = 4
   \]

2. Calculate \(2^4\) mod 53:
   \[
   2^4 = (2^2)^2 = 4^2 = 16
   \]
   \[
   16 \mod 53 = 16
   \]

3. Calculate \(2^8\) mod 53:
   \[
   2^8 = (2^4)^2 = 16^2 = 256
   \]
   \[
   256 \mod 53 = 256 - 4 \times 53 = 256 - 212 = 44
   \]

4. Calculate \(2^{16}\) mod 53:
   \[
   2^{16} = (2^8)^2 = 44^2 = 1936
   \]
   \[
   1936 \mod 53 = 1936 - 36 \times 53 = 1936 - 1908 = 28
   \]

5. Finally, calculate \(2^{17} \mod 53\):
   \[
   2^{17} = 2^{16} \times 2 = 28 \times 2 = 56
   \]
   \[
   56 \mod 53 = 56 - 53 = 3
   \]

So, \(2^{17} \mod 53 = 3\).

Would you like more details or have any questions? 

Here are some related questions:
1. What is \(3^{17} \mod 53\)?
2. How does Fermat's Little Theorem simplify modular exponentiation?
3. How to calculate \(2^{100} \mod 53\)?
4. What is the general approach to finding large powers modulo a prime?
5. How does modular arithmetic apply to cryptography?
6. Can you explain the Chinese Remainder Theorem?
7. What is the significance of Euler's Theorem in number theory?
8. How would you compute \(2^{17} \mod 60\)?

**Tip:** When dealing with large exponents in modular arithmetic, breaking down the problem using successive squaring helps in simplifying the calculations efficiently.

Learn how to calculate 2 raised to the power of 17 modulo 53 using the method of successive squaring. Follow a detailed step-by-step solution with explanations on modular arithmetic and Fermat's Little Theorem. Suitable for advanced high school students and enthusiasts of number theory.

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