Question 01 (f)
Short answer questions: Each question carries 6 marks, your answers should quote the results being used and show your work.
Define a Carmichael number. Use the necessary and sufficient condition for a number to be a Carmichael number to show that 561 is a Carmichael number.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Recall Korselt's criterion which states that a positive composite integer n is a Carmichael number if and only if n is square free and for all prime divisors p of n, it is true that
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
A Carmichael number is a composite number n which satisfies
for all integers with .
For 561, we use Korselt's criterion which states that for each prime dividing n, we must check that . Since
we check that
valid since . Thus 561 is a Carmichael number.