Final answer:
Fermat's little theorem is used in mathematics to evaluate expressions with large exponents modulo a prime number. By stating that a^(p-1) is congruent to 1 mod p for a prime p, it allows for simplification of calculations in modular arithmetic. It's also important to compare outcomes using various methods for verification.
Step-by-step explanation:
Using Fermat's Little Theorem in Evaluations
Using Fermat's little theorem for evaluation is a valuable technique in mathematics, especially when dealing with large exponents in modular arithmetic. The theorem states that if 'p' is a prime number, then for any integer 'a' such that a is not divisible by 'p', it follows that ap-1 ≡ 1 (mod p). This can drastically simplify calculations.
For example, consider evaluating 512 mod 13. Since 13 is prime, we can use Fermat's little theorem which implies that 512 ≡ 1 (mod 13), because 12 is one less than 13. However, if we needed to find 515 mod 13, we would first recognize that 512 * 53 mod 13, and we can use the theorem for the 512 part and then multiply the result by 53 mod 13 to get our answer.
To ensure correctness, comparing outcomes through different methods, such as direct computation, calculators, or iterative experimentation is encouraged. This technique reassures students that they understand the underlying principles and can apply them correctly in various situations, fostering a sense of mastery over mathematical concepts.