Lecture 2

Question 1

Definition of a prime number

Answer 1

Let p>1 be an integer, then p is prime if it’s only positive divisors are 1 and itself.

Question 2

Theorem (prime power factorization)

Answer 2

Let n>1 be an integer. Then n can be factored into prime powers uniquely

Question 3

Formula for the gdc and lcm

Answer 3

m = p_1^e_1 x p_2^e_2 x … x p_r^e_r
n = p_1^f_1 x p_2^f_2 x … x p_r^f_r

Question 4

Consequence of the gcd and lcm formulas

Answer 4

For any m,n which are not zero. Then m x n = gcd(m,n) x lcm(m,n)

Question 5

Euler-Totient function

Answer 5

Let an integer n be greater than or equal to 1. Then the Euler-Totient function of n, is given by phi(n)=number of elements {0<=a<n: gcd(a,n) = 1}

Question 6

Example of the Euler-Totient

Answer 6

phi(5) = 4 because gcd(1,5) = 1, gcd(2,5)=1, gcd(3,5)=1 and gcd(4,5)=1.

Question 7

Division Algorithm

Answer 7

If a and b are integers and b does not equal zero, there are unique integers q and r s.t.
a=q x b + r, 0<=r<|b|
q=quotient of a/b and r is the remainder

Question 8

Definition of a group

Answer 8

A group is a set G with a binary operation * s.t
1) * is associative: (ab)c = a(bc)
2) * has an identity element, denoted e: ae = ea=a
3) * admits inverses: for each a in G there is an a^-1 s.t. a * a^-1 = a^-1 * a = e

Question 9

Example of a group

Answer 9

The set Z (the integers) and the operation + (addition)