To apply **Fermat's Little Theorem** to the expression \(30^3 \mod p\), we first need to understand what Fermat's Little Theorem states:

**Fermat's Little Theorem**: If \(p\) is a prime number and \(a\) is an integer such that \(a\) is not divisible by \(p\), then:

\[
a^{p-1} \equiv 1 \pmod{p}
\]

This can be rewritten for any exponent \(k\) as:

\[
a^k \equiv a^{k \mod (p-1)} \pmod{p}
\]

### Calculation of \(30^3\) using Fermat's Little Theorem:

Let's apply this theorem to \(30^3\) modulo a prime \(p\). Since \(30\) is not divisible by \(p\), Fermat's Little Theorem is applicable. 

For example, if \(p = 31\) (which is a prime), then:

1. According to Fermat's Little Theorem:
   \[
   30^{30} \equiv 1 \pmod{31}
   \]

2. Since we want to find \(30^3 \mod 31\), we calculate directly:
   \[
   30^3 = 27000
   \]

3. Compute modulo 31:
   \[
   27000 \mod 31 = 27000 \div 31 = 870 \text{ remainder } 30
   \]

   Thus:
   \[
   30^3 \equiv 30 \pmod{31}
   \]

Therefore, \(30^3 \equiv 30 \pmod{31}\).

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Would you like more details or have any other questions?

Here are 5 related questions you might be interested in:
1. How can Fermat's Little Theorem be used to simplify large exponentiation problems?
2. What is the generalization of Fermat's Little Theorem (Euler's Theorem)?
3. How would you apply Fermat's Little Theorem if the base \(a\) is divisible by \(p\)?
4. Can Fermat's Little Theorem be used with composite numbers?
5. How does Fermat's Little Theorem relate to modular inverses?

**Tip:** When dealing with large exponents modulo a prime, break down the exponent using Fermat's Little Theorem to reduce the calculations.

Learn how to apply Fermat's Little Theorem to compute 30^3 modulo a prime number. Detailed step-by-step solution provided, including the theorem's application and calculation steps. Suitable for advanced high school students and those interested in number theory and modular arithmetic.

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