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Giveu the following primne decompositions: 0 = M p" 6 = MF PR I; P%. 9 =MNP I-M2 14, Express the &; in terms of the &; and the &; if < = a 6 Tha...

Question

Giveu the following primne decompositions: 0 = M p" 6 = MF PR I; P%. 9 =MNP I-M2 14, Express the &; in terms of the &; and the &; if < = a 6 That is, coplete the following: 3=0.6 iff (Vi eN+) 0 ( Express the Yi in terms of the &; and the B; if g = gcd(a,b) . That is, complete the following: 4 = ged(a.6) iff (Vi eNt)( Ji (C) Express the A, in terms of the 0j AId the B; if [ = lem(a,b) That is, cowplete the followiug: I = lcm(a.6) iff (Vi eN )( 4, Complete the followiug: an

Giveu the following primne decompositions: 0 = M p" 6 = MF PR I; P%. 9 =MNP I-M2 14, Express the &; in terms of the &; and the &; if < = a 6 That is, coplete the following: 3=0.6 iff (Vi eN+) 0 ( Express the Yi in terms of the &; and the B; if g = gcd(a,b) . That is, complete the following: 4 = ged(a.6) iff (Vi eNt)( Ji (C) Express the A, in terms of the 0j AId the B; if [ = lem(a,b) That is, cowplete the followiug: I = lcm(a.6) iff (Vi eN )( 4, Complete the followiug: and b are relatively prime if (Vi eN+)( : (e) Complete the following: 0 aId 6 are relatively prime iff (Bi € N+)( Note: Responses need not be formal aS long A5 terminology and notation usel is clear.



Answers

a) Use Exercise 48 to show that every integer of the form $(6 m+1)(12 m+1)(18 m+1),$ where $m$ is a positive integer and $6 m+1,12 m+1,$ and $18 m+1$ are all primes, is a Carmichael number.
b) Use part (a) to show that $172,947,529$ is a Carmichael number.


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