# In this worksheet, we will test to see if # 172081 is a Carmichael number by applying Korselt's criterion # and only a few basic commands (factor and Mod) # Korselt: We need to check that (1) n is composite and squarefree # and (2) that for every prime p dividing n, that (p-1) divides (n-1) # To check (1), we look at the factorization of n: n=172081 factor(n)

7 * 13 * 31 * 61

# Hence (1) is satisfied: we can see that every factor is only to the 1st power, so n is squarefree # To check condition (2), we will show (p-1) divides (n-1) by checking that (n-1) is 0 modulo (p-1) m=n-1 Mod(m, 6)

0

Mod(m,12)

0

Mod(m,30)

0

Mod(m,60)

0

# Hence (2) is satisfied: p-1 does indeed divide n-1 for all primes p dividing n # We conclude that 172080 is a Carmichael number.

# In this worksheet, we will test to see if # 172081 is a Carmichael number by applying Korselt's criterion # and only a few basic commands (factor and Mod) # Korselt: We need to check that (1) n is composite and squarefree # and (2) that for every prime p dividing n, that (p-1) divides (n-1) # To check (1), we look at the factorization of n: n=172081 factor(n)

7 * 13 * 31 * 61

# Hence (1) is satisfied: we can see that every factor is only to the 1st power, so n is squarefree # To check condition (2), we will show (p-1) divides (n-1) by checking that (n-1) is 0 modulo (p-1) m=n-1 Mod(m, 6)

0

Mod(m,12)

0

Mod(m,30)

0

Mod(m,60)

0

# Hence (2) is satisfied: p-1 does indeed divide n-1 for all primes p dividing n # We conclude that 172080 is a Carmichael number.