30h - Discrete Mathematics Primes 30-2 Integers"God made...

Introduction
Primes
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics - Primes 30-2
Integers
“ God made the integers; all else is the work of man ”
Leopold Kroenecker
Discrete Mathematics - Primes 30-3
Most of useful properties of integers are related to division
If a and b are integers with a ≠ 0, we say that a
divides
b if
there is an integer c such that b = ac.
When a divides b we say that a is a
divisor
(
factor
) of b, and
that b is a
multiple
of a.
The notation a | b denotes that a divides b. We write a | b when
a does not divide b
Example. Let n and d be positive integers. How many positive
integers not exceeding n are divisible by d?
The numbers in question have the form dk, where k is a positive
integer and 0 < dk ≤ n. Therefore, 0 < k ≤ n/d. Thus the answer is
 n/d 
Division
∕
Discrete Mathematics - Primes 30-4
Properties of Divisibility
Let a, b, and c be integers. Then
(i) if a | b and a | c, then a | (b + c);
(ii) if a | b, then a | bc for all integers c;
(iii) if a | b and b | c, then a | c.
Proof.
(i) Suppose a | b and a | c. This means that there are k and m
such that b = ak and c = am.
Then b + c = ak + am = a(k + m), and a divides b + c.
Discrete Mathematics - Primes 30-5
Properties of Divisibility (cntd)
If a, b, and c are integers such that a | b and a | c, then
a | mb + nc whenever m and n are integers.
Proof.
By part (ii) it follows that a | mb and a | nc.
By part (i) it follows that a | mb + nc.
If a | b and b | a, then a = ± b.
Proof.
Suppose that a | b and b | a. Then b = ak and a = bm for some
integers k and m.
Therefore a = bm = akm, which is possible only if k,m = ± 1.
Discrete Mathematics - Primes 30-6
The Division Algorithm
Theorem (The division algorithm)
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
d is called the
divisor
, a is called the