Although ancient Chinese mathematicians did good work coming up with their remainder theorem, they did not always get it right. They had a test for primality. The test said that \(n\) is prime if and only if \(n\) divides \(\left(2^{n}-2ight)\).

a. Give an example that satisfies the condition using an odd prime.

b. The condition is obviously true for \(n=2\). Prove that the condition is true if \(n\) is an odd prime (proving the if condition)

c. Give an example of an odd \(n\) that is not prime and that does not satisfy the condition. You can do this with nonprime numbers up to a very large value. This misled the Chinese mathematicians into thinking that if the condition is true then \(n\) is prime.

d. Unfortunately, the ancient Chinese never tried \(n=341\), which is nonprime \((341=11 \times 31)\), yet 341 divides \(2^{341}-2\) without remainder. Demonstrate that \(2341 \equiv 2(\bmod 341)(\) disproving the only if condition). Hint: It is not necessary to calculate \(2^{341}\); play around with the congruences instead.