Download If a and b are integers, b 0 then a mod b is

GCD and related concepts
If a and b are integers, a  0 then a divides b iff
Notation: a | b
The greatest common divisor of two integers a and b
Example: gcd(18, 24)
If gcd(a, b) = 1, then a and b are relatively prime.
The least common mulltiple of integers a and b is
Example: lcm(18, 24)
Using prime factorization to find the gcd and the lcm:
Division Algorithm and the mod function
Theorem: Let a be an integer and d a positive integer. Then
there exist integers q and r such that a = dq + r, 0  r < d.
If a and b are integers, b  0 then a mod b is
Congruence mod m
If a and b are integers and m is a positive integer > 1 then a is
congruent to b modulo m iff
Notation: a  b (mod m)
The Euclidean Algorithm
Finding the greatest common divisor of a and b
Notation: gcd(a, b)
find gcd (662, 414)
find gcd(330, 156)
662 =
330 =
function gcd(a, b: positive integers)
xa
yb
while y  0
r  x mod y
xy
yr
return(x)
//Note: this differs from the text
Section 2.4 Recursion and Recurrence Relations
Two parts of a recursive definition:
1.
2.
recursively defined sequences
Each of the examples below is a recurrence relation.
Example 1: S(0) = 3 and S(n) = 2S(n – 1) + 3 for n > 0
Example 2: S(0) = 1, S(n) = 3S(n – 1) for n > 1.
Example 3: S(0) = 1, S(n) = nS(n – 1) for n > 0.
Example 4: (Example 30 p. 121) The Fibonacci sequence
The rabbit problem
A pair of young rabbits is placed on an island. A pair of rabbits
does not breed until they are 2 months old. After that, each pair
of rabbits produces another pair each month. Find the number of
rabbits after n months.
Month
breeding
pairs
young
pairs
1
2
3
4
5
Closed form
Example 6: Square Numbers
total pairs
Example 7: Triangular numbers
Recursively Defined Sets
Recursive definition of the set of powers of 3
1.
2.
Recursive definition of S = { …, -10, -5, 0, 5, 10, … }
An alphabet 
A string is
String Terminology
A bit string is
Recursive definition of the set of bit strings B
1.
2.
*--
1.
2.
3.
Concatenation
Examples and properties:
More String Examples
A palindrome is a string
1.
2.
3.
Recurrence relation for S(n) where S(n) is the number of bit
strings of length n that do not contain 11 as a substring.