Let n be a Carmichael number and let p be a prime number that divides n.(a) Finish the pro...

Let n be a Carmichael number and let p be a prime number that divides n.

(a) Finish the proof of Korselt’s Criterion by proving that p − 1 divides n − 1. [Hint. We will prove in Chapter 28 that for every prime p there is a number g whose powers g, g2, g3, . . . , gp−1 are all different modulo p. (Such a number is called a primitive root.) Try putting a = g into the Carmichael congruence an ≡ a (mod n).]

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(b) Prove that p − 1 actually divides the smaller number