2. Discrete Mathematics
Chapter 4
Number Theory

3. Chapter 4
Number Theory
Integers
Integers are whole numbers, without any fractional or decimal components. Example: 8, 21, 8765, –34, 0.
They are in opposite to real numbers, which posses decimal components. Example: 8.0, 34.25, 0.02,

4. Chapter 4
Number Theory
Division of Integers
Suppose a and b are integers, a  0, then a divides b without remaining if there exists an integer c such that b = ac.
Notation: a | b if b = ac, c  Z and a  0.
Example:
4 | 12 because 12/4 = 3 (integer) or 12 = 4  3.
4 | 13 because 13/4 = 3.25 (not integer).

5. Chapter 4
Number Theory
Euclidean Theorem
Suppose m and n are integers, n > 0. If m is divided by n then there exists a unique integer q (quotient) and r (remainder), such that
m = nq + r
where 0  r < n.
m is called dividend, while d is called divisor.
Example:
1987/97 = 20, remaining 47
1987 = 97
25/7 = 3, remaining 4
25 = 73 + 4
–25/7 = –4, remaining 3
–25 = 7(–4) + 3
But not –25 = 7(–3) – 4, because the remainder will be r = –4, while the condition is 0  r < n)

6. Greatest Common Divisor (GCD)
Chapter 4
Number Theory
Greatest Common Divisor (GCD)
Suppose a and b are non-zero integers. The Greatest Common Divisor (GCD) of a and b is the greatest possible integer d such that d | a and d | b.
In this case, it can be written as GCD(a,b) = d.
Example: Determine GCD(45,36) !
Divisors of 45: 1, 3, 5, 9, 15, 45.
Divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
Common divisors of 45 and 36 are 1, 3, 9.
For the enumeration above, it can be concluded that GCD(45,36) = 9.

7. Greatest Common Divisor (GCD)
Chapter 4
Number Theory
Greatest Common Divisor (GCD)
Suppose m and n are integer, n > 0, such that m = nq + r, 0  r < n. Then GCD(m,n) = GCD(n,r).
Example:
Take the value m = 66, n = 18,
66 = 183 + 12
then GCD(66,18) = GCD(18,12) = 6.

8. Chapter 4
Number Theory
Euclidean Algorithm
Euclid (around 300 BC) is a Greek mathematician, who wrote this algorithm in his book titled, “Element.”
The objective of this algorithm is to find the GCD of two integers.

9. Chapter 4
Number Theory
Euclidean Algorithm
If m and n are non-negative integers where m  n, and suppose r0 = m and r1 = n.
Perform the following divisions in sequence to obtain:
r0 = r1q1 + r2 0  r2  r1,
r1 = r2q2 + r3 0  r3  r2,
ri–2 = ri–1qi–1 + ri 0  ri  ri–1,
ri–1 = riqi + 0
. . .
. . .
According to the statement on the previous section,
GCD(m,n) = GCD(r0,r1) = GCD(r1,r2) = … =
GCD(ri–2,ri–1) = GCD(ri–1,ri) = GCD(ri,0) = ri
Thus, GCD of m and n is the last non-zero remainder of the above sequence of disions, namely ri.

10. Chapter 4
Number Theory
Euclidean Algorithm
Given two non-negative integers m and n (m  n), the following Euclidean Algorithm will find the greatest common divisor of m and n.
Step 1: If n = 0 then m is the GCD(m,n); Finish.
If n  0, proceed to Step 2.
Step 2: Divide m with n and obtain r as the remainder.
Step 3: Replace m with n, and n with r, then loop back to Step 1.

11. Euclidean Algorithm Example:
Chapter 4
Number Theory
Euclidean Algorithm
Example:
Take m = 80, n = 12, so the condition that m  n is fulfilled.
80 = 126 + 8
12 = 81 + 4
8 = 42 + 0
n = 0  m = 4 is the last non-zero remainder
GCD(80,12) = 4; Finish.

12. Chapter 4
Number Theory
Linear Combination
GCD(a,b) can be expressed as a linear combination of a and b with the multiplying coefficients that can be freely chosen.
Example: GCD(80,12) = 4, then 4 = (–1)80 + (7)12, where –1 and 7 are coefficients that can be freely chosen.
Suppose a and b are positive integers, then there exist integers m and n such that GCD(a,b) = ma + nb.

13. Chapter 4
Number Theory
Linear Combination
Example: Express GCD(312,70) = 2 as the linear combination of 312 and 70!
Applying Euclidean Algorithm:
312 = 4 (1)
70 = 2 (2)
32 = 56 + 2 (3)
6 = 32 + 0 (4)
Thus, GDC(312,70) = 2
Rearrange (3) to
2 = 32 – 56 (5)
Rearrange (2) to
6 = 70 – 232 (6)
Insert (6) to (5) so that
2 = 32 – 5(70 – 232)
= 132 – 5 32
= 1132 – 570 (7)
Rearrange (1) to
32 = 312 – 470 (8)
Insert(8) to (7) so that
2 = 1132 – 570
= 11(312 – 470) – 570
= 11312 – 4970
Thus, GCD(312, 70) = 2

14. Chapter 4
Number Theory
Modulo Arithmetics
Suppose a is an arbitrary integer and m is a positive integer, then
a mod m yields the remainder if a is divided by m
a mod m = r such that a = mq + r, with 0  r < m
The result of modulo m lies within the set {0,1,2,…,m–1}

15. Chapter 4
Number Theory
Congruence
Examine that 38 mod 5 = 3 and 13 mod 5 = 3, then it can be written that 38  13 (mod 5).
Pronounce: 38 is congruent with 13 in modulo 5.
Suppose a and b are integers and m > 0. If m divides a – b without remainder, then a  b (mod m).
If a is not congruent with b in modulo m, then it is written as a  b (mod m).

16. Congruence Example: 17  2 (mod 3)
Chapter 4
Number Theory
Congruence
Example:
17  2 (mod 3)
 3 divides 17–2 = 15 without remainder
–7  15 (mod 11)
 11 divides –7–15 = –22 without remainder
12  2 (mod 7)
 7 cannot divide 12–2 = 10
–7  15 (mod 3)
 3 cannot divide –7–15 = –22

17. Congruence a  b (mod m) can be written as a = b + km (k integer).
Chapter 4
Number Theory
Congruence
a  b (mod m) can be written as a = b + km (k integer).
Example:
17  2 (mod 3)  17 = 2 + 53
–7  15 (mod 11)  –7 = 15 + (–2)11
a mod m = r can also be written as a  r (mod m).
Example:
23 mod 5 = 3  23  3 (mod 5)
14 mod 8 = 6  14  6 (mod 8)
–41 mod 9 = 4  –41  4 (mod 9)
–39 mod 13 = 0  –39  0 (mod 13)

18. Congruence Theorem Suppose m is a positive integer.
Chapter 4
Number Theory
Congruence Theorem
Suppose m is a positive integer.
If a  b (mod m) and c is an arbitrary integer, then
(a + c)  (b + c) (mod m)
ac  bc (mod m)
ap  bp (mod m) , p non-negative
If a  b (mod m) and c  d (mod m), then
(a + c)  (b + d) (mod m)
ac  bd (mod m)

19. Chapter 4
Number Theory
Congruence
Example: Suppose 17  2 (mod 3) and 10  4 (mod 3), then according to the Congruence Theorem,
 (mod 3)  22  7 (mod 3)
175  25 (mod 3)  85  10 (mod 3)
 (mod 3)  27  6 (mod 3)
1710  24 (mod 3)  170  8 (mod 3)

20. Chapter 4
Number Theory
Prime Numbers
A positive integer p (p > 1) is called a prime number if its divisors are only 1 and p.
For example, 23 is a prime number, because it can only be divided by 1 and 23 to get no remainder.
Numbers which are not prime numbers are called composite numbers.
For example, 20 is a composite number, because 20 is divisible by 2, 4, 5, and 10, besides by 1 and 20 itself.

21. Chapter 4
Number Theory
Relatively Prime
Two integers a and b are said to be relatively prime if they do not have any common factors other than 1, or, GCD(a,b) = 1.
Example:
20 and 3 are relatively prime, since GCD(20,3) = 1.
7 and 11 are relatively prime, since GCD(7,11) = 1.
20 and 5 are not relatively prime, since GCD(20,5) = 5 ≠ 1.

22. Chapter 4
Number Theory
Relatively Prime
If a and b are relatively prime, then there exist integers m and n such that ma + nb = 1.
Example:
20 and 3 are relatively prime because GCD(20,3) =1, so that it can be written that 220 + (–13)3 = 1 (m = 2, n = –13).
20 and 5 are not relatively prime because GCD(20,5) ≠ 1, and thus 20 and 5 cannot be written in the form of m20 + n5 = 1.

23. Linear Congruence The linear congruence is in the form of:
Chapter 4
Number Theory
Linear Congruence
The linear congruence is in the form of:
ax  b (mod m),
where m > 0, a and b are arbitrary integers, and x is any integer.
The solution can be found in the way:
ax = b + km  x = (b + km) / a
Try each value of k = 0, 1, 2, … and k = –1, –2, … that delivers integer value for x.

24. Chapter 4
Number Theory
Linear Congruence
Example: Determine the solutions for 4x  3 (mod 9) !
4x  3 (mod 9)  x = (3 + k9 ) / 4
k = 0  x = (3 + 09) / 4 = 3/4  not a solution
k = 1  x = (3 + 19) / 4 = 3  a solution
k = 2  x = (3 + 29) / 4 = 21/4  not a solution
k = 3, k =  no solution
k = 5  x = (3 + 59) / 4 = 12  a solution …
k = –1  x = (3 – 19) / 4 = –6/4  not a solution
k = –2  x = (3 – 29) / 4 = –15/4  not a solution
k = –3  x = (3 – 39) / 4 = –6  a solution
k = –7  x = (3 – 79) / 4 = –15  a solution
The set of solutions is: {3, 12, …, –6, –15, …}.

25. Chapter 4
Number Theory
Linear Congruence
Example: Determine the solutions for 2x  3 (mod 4) !
2x  3 (mod 4)  x = (3 + k4 ) / 2
Because k4 is always an even number, then 3 + k4 will always be an odd number.
If an odd number is divided by 2, then the result will be a decimal number (never be an integer).
Thus, there is no value of x that can be the solution of 2x  3 (mod 4).

26. Chapter 4
Number Theory
Homework 6
Determine GCD(216,88) and express the GCD as a linear combination of 216 and 88.

27. Chapter 4
Number Theory
Homework 6A
Determine GCD(74,16) and express the GCD as a linear combination of 74 and 16.
Determine the solutions for 5x  7 (mod 11) !