Note that 561 is not prime, because 561 = 3.11.17. Interestingly, 561 satisfies a curious property that a561 = a (mod 561) holds for every integer a, which makes it 561 an example of a Carmichael number. The goal of this problem is prove this assertion about the number 561, and also explore a bit more on how to find more examples of such numbers (in the last part of the problem). (a) Explain why the following congruences a = a (mod 3) and a = a (mod 11) and a = a (mod 17) 1 are true for every integer a. (b) Use the previous part to conclude that the following congruences 561 561 a561 = a (mod 3) and a = a (mod 11) and a = a (mod 17) are true for every integer a. 561 (c) Explain how the previous three congruences allow us to deduce that a for every integer a. (d) Lookup Korselt's criterion in a book or online. Write a brief description of how it works, and use it to show that 29341 = 13.37-61 and 172947529=307-613.919 are Carmichael numbers. = a (mod 561)