12.1.5. Applications of Fermat's Little Theorem

Try these first

1. 1.

   What does Fermat's Little Theorem state?

   Hint

   Focus on the conditions involving **p** and **a**.

2. 2.

   Give an example of a number that is co-prime to 7.

   Hint

   Think about multiples.

3. 3.

   What does Fermat's Little Theorem allow us to conclude if b^(p-1) ≡ 1 (mod p)?

   • **p is prime**
   • **p is composite**
   • **a is prime**

   Hint

   Remember valid conditions of the theorem.

4. 4.

   Is the statement 'All Carmichael numbers are prime' true?

   • True
   • False

   Hint

   Recall the definition of a Carmichael number.

5. 5.

   Using Fermat's Little Theorem, prove that a^(p-1) ≡ 1 (mod p) holds for large p and any a co-prime to it. Provide an example.

   Hint

   Select a prime larger than 10.

6. 6.

   Explore the implications of Carmichael numbers. Investigate if any Carmichael numbers exist and discuss their significance.

   Hint

   Use online mathematical databases or number theory resources.