2. “The laws of nature are
but the mathematical
thoughts of God ”
-Euclid
2
3. DIVISIBILITY
Definition: 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. If a divides b, we also say
that a is a divisor or factor of b and that b is a
multiple of a.
If a divides b, we write ȁ
𝑎 𝑏 , while if a does
not divide b, we write a ∤ 𝑏
4. DIVISIBILITY
The notation a | b to mean that a divides b. Be
careful not to confuse “a | b” with “a/b” or
“a÷b”. The notation “a | b” is read “a divides
b”, which is a statement — a complete
sentence which could be either true or false.
On the other hand, “a ÷ b” is read “a divided
by b”. This is an expression, not a complete
sentence. Compare “6 divides 18” with “18
divided by 6” and be sure you understand the
difference
5. DIVISIBILITY
The properties in the next proposition are easy
consequences of the definition of divisibility;
Proposition.
Every nonzero number divides 0.
1 divides everything. So does −1.
Every nonzero number is divisible by itself.
6. Basic theorem of divisibility
1. If a|b and a|c, then a|(b + c).
2. If a|b, then a|(bc).
3. If a|b and b|c, then a|c.
6
7. Basic theorem of divisibility
THEOREM #1:If a|b and a|c, then a|(b + c).
3 | 6 and 3 | 9, so 3 | 15.
5| 35 and 5 | 25, so 5 | 60.
9 | 27 and 8 | 81, so 9 | 99.
7
8. Basic theorem of divisibility
THEOREM #2: If a|b, then a|(bc).
5 | 10, so 5 | 20, 5 | 30.
15 | 45, so 15 | 60, 5 | 2700.
11 | 66, so 11 | 121, 11 | 7986.
8
9. Basic theorem of divisibility
THEOREM #3 :If a|b and b|c, then a|c.
4 | 8 and 8 | 24, so 4 | 24
12 | 36 and 36 | 72, so 12 | 72
24 | 120 and 120 | 720, so 24 | 720
9
10. Complete the following statement using the theorem #1
15 | 45 and 15 | 30, so 3 | ?.
75
21| 63 and 21 | 441, so 21 | ?.
504
212 | 1060 and 212 | 2332, so 212 | ?.
3,392
10
11. Complete the following statement using the theorem #2
41 | 246, so 41 | 451, 41 | ?.
110,946
231 | 2772, so 231 | 4851, 231 | ?.
13,446,972
406 | 3248, so 406 | 5239, 406 | ?.
17,016,272
11
12. Complete the following statement using theorem #3
51 | 561 and 561 | 4488, so ?
51 | 4488
721 | 2884 and 2884 | 31724, so ?
721 | 31,724
1605 | 17,655 and 17655 | 88275, so ?
1,605 | 88,275
12
13. The Language in Which Mathematics Is Done
Even when talking about day to day life, it sometimes happens that one has a hard time
finding the words to express one’s thoughts. Furthermore, even when one believes that
one has been totally clear, it sometimes happens that there is a vagueness in one’s words
that results in the listener understanding something completely different from what the
speaker intended. Even the simplest mathematical reasoning is usually more complicated
than the most complicated things one attempts to discuss in everyday life. For this
reason, mathematics has developed an extremely specialized (and often stylized)
language. To see the value of this, one only has to think of how difficult it would be to
solve even the simple quadratic equation x2 − 5x = 14 if one’s calculation had to be done
without symbols, using only ordinary language.
“Suppose that a quantity has the property that when five times that quantity has the
property that when five times that quantity is subtracted from the product of the
quantity times itself then the result is 14. Determine the possible values of this quantity.”
13
14. The Language in Which Mathematics Is Done
During the middle ages, algebraic calculations were in fact done in this fashion — except
that the language was Latin, not English. But progress beyond the level of simple algebra
only became possible with the introduction of modern algebraic notation — which the West
learned from the Arabs. By combining symbolic notation with very precisely defined
concepts and a very formalized language, any mathematical proof can be expressed in a
way that can be understood by any “mathematically mature” reader. One is not dependent
on the reader’s ability to “see what is meant.”
14
15. DIVISION ALGORITHM
THEOREM
If a and b are integers such that b>0, then there
are unique integers q and r such that
a =bq + r with 0 ≤ 𝑟 < 𝑏.
In this theorem, q is the quotient and r is the
remainder, a the dividend and b the divisor.
We note that a is divisible by b if and only if
the remainder in the division algorithm is 0.
16. DIVISION ALGORITHM
THEOREM
• The Division Algorithm says that an integer can be divided by
another (nonzero) integer, with a unique quotient and
remainder.
18. DIVISION ALGORITHM
THEOREM
1. If a = 82 and b = 7, then 82 = (7)(11) + 5:
2. If a = 1 and b = 20, then 1 = (20)(0) + 1:
3. If a = 0 and b = 15, then 0 = (15)(0) + 0:
19. DIVISION ALGORITHM
THEOREM
a= -36, b=5
−36
5
= −7.2
round down, q=-8
-36=5(-8)+r
Find r:
Rewrite the equation
a=bq+r
r=-bq+a
r=-(5(-8))+(-36)
r=40-36
r=4
By the division
algorithm,
-36=5(-8)+4
20. DIVISION ALGORITHM
THEOREM
a= -413, b=3
−413
3
= −137.6
round down, q=-138
-413=3(-138)+r
Find r:
Rewrite the equation
a=bq+r
r=-bq+a
r=-(3(-138))+(-413)
r=414-413
r=1
By the division
algorithm,
-413=3(-138)+1
21. DIVISION ALGORITHM
THEOREM
a= -389, b=16
−389
16
= −24.31
round down, q=-25
-389=16(-25)+r
Find r:
Rewrite the equation
a=bq+r
r=-bq+a
r=-(16(-25))+(-389)
r=400-389
r=11
By the division
algorithm,
-389=16(-25)+11
22. DIVISION ALGORITHM
THEOREM
1. If a = -17 and b = 3,
then q = -6 and r = 1. −17 = 3 −6 + 1
2. If a = 18 and b = 6,
then q = 3 and r = 0. 18 = 6 3 + 0
3. If a = −79 and b = 9,
then q=-9 and r=2 −79 = 9 −9 + 2
23. Prime numbers
23
Prime numbers are the building blocks of arithmetic. At the moment there
are no efficient methods (algorithms) known that will determine whether a
given integer is prime or find its prime factors. This fact is the basis behind
many of the cryptosystems currently in use. One problem is that there is no
known procedure that will generate prime numbers, even recursively. In fact,
there are many things about prime numbers that we don’t know. For example,
there is a conjecture, known as Goldbach’s Conjecture, that there are
infinitely many prime pairs, that is, consecutive odd prime numbers, such as
5 and 7, or 41 and 43, which no one so far has been able to prove or disprove.
As the next theorem illustrates, it is possible, however, to prove that there
are infinitely many prime numbers. Its proof, attributed to Euclid, is one of
the most elegant in all of mathematics.
24. Prime numbers
24
A prime is an integer with precisely two positive integer divisors. In
the past three centuries, the mathematicians have devoted countless
hours in exploring the world of primes. They have discovered many
fascinating properties, formulated diverse conjectures, and proved
interesting and surprising results.
25. Prime numbers
25
A prime is a positive integer greater than 1 that is divisible by no
positive integers other than 1 and itself.
e.g. 1,2,3,5,7,11, 179
27. GREATEST COMMON DIVISOR
27
The greatest common divisor (gcd) of two or more non- zero integers
is the largest positive integer that divides the numbers without a
remainder.
28. GREATEST COMMON DIVISOR
28
The common divisors of 36 and 60 are 1, 2, 3, 4, 6, 12.
The greatest common divisor gcd(36,60) = 12.
34. 34
Euclid
Euclid is often referred to as the “Father of
Geometry”, and he wrote perhaps the
most important and successful
mathematical textbook of all time, the
“Stoicheion” or “Elements”, which
represents the culmination of the
mathematical revolution which had taken
place in Greece up to that time. He also
wrote works on the division of geometrical
figures into into parts in given ratios, on
catoptrics (the mathematical theory of
mirrors and reflection), and on spherical
astronomy (the determination of the
location of objects on the “celestial
sphere”), as well as important texts on
optics and music.
35. 35
> The well known Euclidean algorithm finds the greatest
common divisor of two numbers using only
elementary mathematical operations - division and
subtraction
Euclidean Algorithm
36. 36
Euclidean Algorithm
> A divisor of a number a is an integer that divides it without
remainder
> For example the divisors of 12 are 1, 2, 3, 4, 6 and 12
> The divisors of 18 are 1, 2, 3, 6, 9 and 18.
37. 37
Euclidean Algorithm
> The greatest common divisor, or GCD, of two numbers is
the largest divisor that is common to both of them.
> For example GCD(12, 18) is the largest of the divisors
common to both 12 and 18.
38. 38
Euclidean Algorithm
> The greatest common divisor, or GCD, of two numbers is
the largest divisor that is common to both of them.
> For example GCD(12, 18) is the largest of the divisors
common to both 12 and 18.
40. 40
Euclidean Algorithm
> The Euclidean Algorithm to find GCD(a, b) relies upon
replacing one of a or b with the remainder after division.
> Thus the numbers we seek the GCD of are steadily
becoming smaller and smaller. We stop when one of them
becomes 0.
41. 41
Euclidean Algorithm
> Specifically, we assume that a is larger than b. If b is
larger than a, then we swap them around so that a becomes
the old b and b becomes the old a.
> We then look for numbers q and r so that a=bq+r. They
must have the properties that q0 and 0r<b.
> In other words, we seek the largest such q.
42. 42
Euclidean Algorithm
> As examples, consider the following.
> a=12, b=5; 12=5*2+2 so q=2, r=2
> a=24, b=18; 24=18*1+6 so q=1, r=6
> a=30, b=15; 30=15*2+0 so q=2, r=0
> a=27, b=14; 27=14*1+13 so q=1, r=13
> Try the ones on the next slide.
43. 43
Euclidean Algorithm
> Find q and r for the following sets of a and b. The answers
are on the next slide.
> a=28, b=12
> a=50, b=30
> a=35, b=14
> a=100, b=20
45. 45
Euclidean Algorithm
> The algorithm works in the following way.
> Given a and b, we find numbers q and r so that a=bq+r.
> We make sure that q is as large as possible (≥0), and 0≤r<b.
> For example, if a=18, b=12, then we write 18=12*1+6.
46. 46
Euclidean Algorithm
> Once the remainder r has been found we replace a by b and
b by r.
> This relies on the fact that GCD(a,b)=GCD(b,r).
> Hence we repeatedly find r, the remainder after a is divided
by b.
> Then replace a by b and b by r, and keep on in this way until
r=0.
47. 47
Euclidean Algorithm
> Let us look at a graphical interpretation of the Euclidean
algorithm.
> Obviously if p=GCD(a,b) then p|a and p|b, that is to say p
divides both a and b evenly with no remainder.
48. 48
Euclidean Algorithm
> Let us look at a graphical interpretation of the Euclidean
algorithm.
> Obviously if p=GCD(a,b) then p|a and p|b, that is to say p
divides both a and b evenly with no remainder.
49. 49
Show the computations for a=210 and b=45.
• Divide 210 by 45, and get the result 4 with remainder
30, so 210=4·45+30.
• Divide 45 by 30, and get the result 1 with remainder
15, so 45=1·30+15.
• Divide 30 by 15, and get the result 2 with remainder
0, so 30=2·15+0.
• The greatest common divisor of 210 and 45 is 15.
50. 50
Show the computations for a=420 and b=55.
• Divide 420 by 55, and get the result 7 with remainder 35,
so 210=7*55+35.
• Divide 55 by 35, and get the result 1 with remainder 20, so
55=1·35+20.
• Divide 35 by 20, and get the result 1 with remainder 15, so
35=1·20+15.
• Divide 20 by 15 and get the result 1 with remainder 5, so
20= 1 * 15 + 5
• Divide 15 by 5 and get the result 3 with remainder 0, so
15= 3 * 5 + 0
• The greatest common divisor of 420 and 55 is 5.
