Practice Final Remarks On Number Theory (12.5.2) - Introduction to Fermat’s Little Theorem and Primality Testing

Practice: Final Remarks on Number Theory

Lesson 12.5.2Introduction to Fermat’s Little Theorem and Primality Testing

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8 questions

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6 sample questions

Question 1.
State Fermat's Little Theorem in your own words.
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Think about what happens when you raise numbers that fit this requirement.
Question 2.
What is a Carmichael number? Give an example.
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These numbers can fool primality tests!
3Question 3.
Which statement accurately describes Fermat's Little Theorem?
- If p is prime and a is any integer
- then a^(p-1) ≡ 0 (mod p)
- If p is prime and a is co-prime to p
- then a^(p-1) ≡ 1 (mod p)
- Carmichael numbers are prime
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Which option keeps the focus on co-primality?
4Question 4.
True or False: All composite numbers are Carmichael numbers.
- True
- False
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Think about examples of composites that are not Carmichael.
Question 5.
Given the number 561, demonstrate why it is a Carmichael number using Fermat’s Little Theorem for bases 2, 3, and 5.
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Use Fermat's theorem for your calculations.
Question 6.
Construct your primality testing algorithm incorporating Fermat’s theorem, listing steps you would take to strengthen it against Carmichael numbers.
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Think about involving multiple tests!

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