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Calculate 3^274 With formats little theorem And mod 31

To calculate \(3^{274} \mod 31\) using **Fermat's Little Theorem**, let's follow these steps:

### 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:
\[
a^{p-1} \equiv 1 \mod p
\]
In our case, \( a = 3 \) and \( p = 31 \), so:
\[
3^{30} \equiv 1 \mod 31
\]

### Reducing the Exponent Modulo \( 30 \)
Given the exponent \( 274 \), we can reduce it modulo \( 30 \) since \( 3^{30} \equiv 1 \mod 31 \):
\[
274 \mod 30 = 4
\]
Thus, \( 3^{274} \equiv 3^4 \mod 31 \).

### Calculating \( 3^4 \mod 31 \)
Now, compute \( 3^4 \):
\[
3^4 = 81
\]
Finally, find \( 81 \mod 31 \):
\[
81 \mod 31 = 19
\]
So, \( 3^{274} \equiv 19 \mod 31 \).

### Final Answer
\[
3^{274} \equiv 19 \mod 31
\]

Would you like more details or have any questions about this process? 

Here are some related questions you might be interested in:
1. How does Fermat's Little Theorem work in other examples?
2. Can Fermat's Little Theorem be applied to non-prime moduli?
3. What are some real-world applications of Fermat's Little Theorem?
4. How would you calculate \( 3^{274} \mod 31 \) using a different method?
5. What are other useful theorems in modular arithmetic?

**Tip:** Remember, Fermat's Little Theorem is particularly useful when dealing with large exponents in modular arithmetic!

Learn how to calculate 3^274 mod 31 using Fermat's Little Theorem and modular arithmetic. Follow a step-by-step solution reducing the exponent modulo 30 and applying modular arithmetic principles. Suitable for advanced high school students and anyone interested in number theory and modular arithmetic.

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