Mathematics 5160
Exam 2
Name:
Date:
(1) (5pts) State the Fundamental Theorem of Arithmetic.
(2) (5pts) State Wilson’s Theorem.
(3) (5pts) State the definition of the Euler phi-function.
(4) (5pts) State Fermat’s Little Theorem.
(5) (35pts) Determine whether the following statements are always true, sometimes true, or
never true. Justify each answer with an example or explanation.
(a)
Let a and b be positive integers. If there exists integers x and y such that ax+by = d,
then gcd(a, b) = d.
(b)
If a ≡ b mod n and c ≡ d mod n, then ab ≡ cd mod n.
(c)
The positive integer a is its own inverse modulo n if and only if a ≡ 1 mod n or
a ≡ −1 mod n.
(d)
If p is a prime with p ≡ 3 mod 4, then φ(pk ) = (p − 1)k where k ∈ Z+ with k ≥ 2.
(e)
If f (x) is a polynomial of degree n with integral coefficients and p is a prime such that
p does not divide the leading coefficient of f (x), then the congruence f (x) ≡ 0 mod p
has at most n mutually incongruent solutions modulo p
(f)
If a and b are elements of Z∗n , then their product ab is an element of Z∗n .
(g)
If ak ≡ bk mod n and k ≡ j mod n, then aj ≡ bj mod n.
(6) (30pts) Choose three of the following four problems.
(a)
Show that if p is a prime, a is an integer, and p | an , then pn | an .
(b)
Let p be a prime. Prove that (a + b)p ≡ ap + bp mod p.
(c)
Let p be a prime such that p ≡ 1 mod 4. Show that if there exists a solution to
x2 ≡ s mod p, then there exists a solution to x2 ≡ (p − s) mod p.
(d)
Prove that for any positive integer n there exists a string of n consecutive composite
integers.
(7) (5pts) What is the least positive residue of 31547329824 modulo 11?
(8) (5pts) Find all solutions of the system of congruences x ≡ 1 mod 2, x ≡ 3 mod 5, and
x ≡ 4 mod 7.
(9) (5pts) Determine if the equation 42x + 175y = 21 can be solved in the integers. If such a
solution exists, find the appropriate values for x and y.