1. CMSC 203, Section 0401 Discrete Structures Fall 2004 Matt Gaston

2. Integers Ch

3. Division Let a and b be integers with a  0. Then a divides b
if there is an integer c such that b = ac. When a divides
b, a is called a factor of b and b is called a multiple of a.
The notation a | b denotes a divides b.

4. Primes A positive integer p > 1 is called prime if the only
positive factors of p are 1 and p. A positive integer
that is greater than 1 and is not prime is called
composite.

5. The Fundamental Theorem of Arithmetic
Theorem: Every positive integer greater than
1 can be written uniquely as a prime or as the
product of two or more primes where the prime
factors are written in order of nondecreasing size.

6. The Division Algorithm
Theorem: Let a be an integer and d a positive
integer. Then there are unique integers q and r,
with 0  r  d, such that a = dq + r.
In the equality given in the above theorem, a is
called the dividend,d is called the divisor, q is
called the quotient,and r is called the remainder.
This notation is used to express the quotient and
remainder:
q = a div d r = a mod d.

7. Greatest Common Divisor
Let a and b be integers, but not both 0. The largest
integer d such that d | a and d | b is called the
greatest common divisor of a and b. [ gcd(a,b) ]

8. Least Common Multiple Let a and b be positive integers. The largest
integer d such that a | d and b | d is called the
least common multiple of a and b. [ lcm(a,b) ]

9. Modular Arithmetic
If a and b are integers and m is a positive integer,
then a is congruent to b modulo m if m | (a – b).
[ a  b (mod m) ]

10. Congruence Theorems Theorem: Let m be a positive integer. Then
a  b (mod m) if and only if a mod m = b mod m.
Theorem: Let m be a positive integer. Then
a  b (mod m) if and only if there is an integer
k such that a = b + km.
Theorem: Let m be a positive integer. If
a  b (mod m) and c  d (mod m), then
a + c  b + d (mod m) and
ac  bd (mod m).

11. Base b Expansions Theorem: Let b be a positive integer greater than 1.
Then if n is a positive integer, it can be expressed
uniquely in the form
n = akbk + ak-1bk a1b + a0,
where k is a nonnegative integer, a0 , a1 , , ak are
Nonnegative integers less than b.

12. Constructing Base b Expansions
procedure base b expansion (n:positive integer)
q := n
k := 0
while (q  0)
begin
ak := q mod b
q :=  q / b 
k := k + 1
end {the base b expansion of n is (ak-1 ak a1 a0 )b }

13. Addition of Integers procedure add (a,b:positive integers) c := 0
for j := 0 to n - 1
begin
d :=  (aj + bj + c) / 2 
sj := aj + bj + c - 2d
c := d
end
sj := c
{the binary expansion of the sum is (sn sn s0 )2 }

14. Multiplying Integers procedure multiply (a,b:positive integers) c := 0
for j := 0 to n - 1
begin
if bj then cj := a shifted j places
else cj := 0
end
p := 0
for j := 0 to n – 1
p := p + cj
{p is the value of ab }

15. Euclidean Algorithm Lemma: Let a = bq + r, where a, b, q, and r are
integers. Then gcd(a, b) = gcd(b, r)
procedure procedure (a,b:positive integers)
x := a
y := b
while y  0
begin
r := x mod y
x := y
y := r
end { gcd(a, b) is x }