Math 4430
Problem set #3
Solutions
Due Monday, February 6
Exercise 1. Find all solutions to x2 ≡ −1 (mod 65). Now do the same modulo 5 · 7 · 13.
Exercise 2. Prove that 11104 + 1 is a multiple of 17. Hint: use little Fermat.
Exercise 3. If p and q are distinct primes, prove that
pq−1 + q p−1 ≡ 1
(mod pq).
Exercise 4. Let a and b be integers, both relatively prime to 91. Show that
a12 ≡ b12
(mod 91).
Hint: 91 is not prime.
Exercise 5.
(a) Assuming that gcd(m, n) = 1, construct a bijection
f : (Z/mnZ)× → (Z/mZ)× × (Z/nZ)× .
Deduce that ϕ(mn) = ϕ(m)ϕ(n).
(b) If p is prime, prove that ϕ(pk ) = pk − pk−1 .
Exercise 6.
(a) Suppose that p1 , . . . , pr is a list of the distinct prime factors of n. Prove that
1
1
ϕ(n) = n · 1 −
··· 1 −
.
p1
pr
(b) Compute ϕ(59535). Hint: 59535 = 35 · 5 · 72 .
Exercise 7. Recall from class that n ∈ Z+ is a Carmichael number if it is composite and
satisfies
an−1 ≡ 1 (mod n)
whenever gcd(a, n) = 1. Prove that 561 = 3 · 11 · 17 is a Carmichael number.
Exercise 8. Suppose n is a Carmichael number, and p is a prime divisor of n. Let pk be
the largest power of p dividing n.
(a) Prove that (1 + p)n−1 ≡ 1 (mod pk ).
(b) Deduce that k = 1.
In other words: a Carmichael number can have no repeated prime factors.