MAT 312/AMS 351 - Applied Algebra -- Fall 2004

Problems marked with an asterisk (*) are for extra credit.

Section 1.6: Ex 9^*, 10, 12
Find the primes p and q if pq=4,386,607 and φ(pq)=4,382,136. Explain the method you have used.
Compute φ(n) for the following values of n= 10!, 20! and 1001. Explain the method you have used.
Are there any numbers n such that φ(n)=14? Explain!
Find the remainder at division of 3¹⁰⁰⁰ by 35.

Section 2.3: Ex 1, 2 (only c, e, d, f), 4, 5, 6, 7, 9
Let X= {1, 2, 3, 4, 5}. For each part, define a relation R on the set X so that R is
1. reflexive and symmetric, but not transitive.
2. symmetric and transitive, but not reflexive.
3. reflexive, symmetric, transitive, and weakly antisymmetric.
Let M be the relation on the real numbers R defined as follows: for all x, y ∈R, xMy if and only if
x-y
is an integer.
1. Prove that M is an equivalence relation.
2. Describe the equivalence classes of M.

Section 4.1: Ex 1 (only the products π₁π₂, π₂π₃ and π₂π₁), 2, 4, 5, 6
Write the permutations π₁, π₂ and π₄ in the exercise 1 of section 4.1 as a product of disjoint cycles.

Section 4.3: 2, 3, 5, 6, 8
Describe the group of symmetries of a regular hexagon. How many symmetries does it have? What do they look like? What relationships are there among them?

MAT 312/AMS 351 Applied Algebra