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Departamento de
Matemáticas
U1_Bilingüe
I.E.S. MIGUEL HERNÁNDEZ
Alhama de Murcia
Unit 1: Divisibility & Integers
Mathematicians have always been fascinated by numbers—their structure and
their uses. Last year we studied whole numbers, integer numbers and divisibility and
this year we are going to revise it. The structure of the counting numbers—their basic
building blocks—is similar in nature to the concept of all molecules being made up of
atoms of the basic elements. As a molecule of water is formed from two atoms of
hydrogen and one atom of oxygen, H2O, so the number 12 is made up of two factors of
2 and one factor of 3, that is, 12 = 2 · 2 · 3.
Throughout the history of humanity, people have studied different sets of
numbers. Some groups have even attributed magical powers to certain numbers (“lucky
seven”) or groups of numbers because of some special properties. We investigate here
the notion of a “magic square.”
Magic squares are arrangements of the numbers 1–4, or 1–9, or 1–16, or 1–25,
or 1–any number squared. These numbers are written in square arrays that are 2 × 2, or
3 × 3, or 4 × 4, and so on. The numbers are placed so that the sum of each row and of
each column and of each diagonal is the same. Figure below shows an example of a 3 ×
3 magic square, which uses the numbers from 1 to 9 (32).
2 9 4
7 5 3
6 1 8
Row 1
Row 2
Row 3
Diagonal
2 + 9 + 4 = 15
7 + 5 + 3 = 15
6 + 1 + 8 = 15
2 + 5 + 8 = 15
Column 1
Column 2
Column 3
Diagonal
2 + 7 + 6 = 15
9 + 5 + 1 = 15
4 + 3 + 8 = 15
4 + 5 + 6 = 15
In medieval times, some people wore magic squares and used them as
talismans. The talisman wearers considered them powerful enough to provide
protection from evil spirits.
Key Words I
1
2
3
4
5
6
7
8
9
10
whole numbers
integer numbers
block
set
power
arrangement
row
column
even number
composite number
The Sagrada Familia Magic Square
Unit_1_Divisibility_&_Integers
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Departamento de
Matemáticas
U1_Bilingüe
I.E.S. MIGUEL HERNÁNDEZ
Alhama de Murcia
1.1: Multiples and factors
> Multiples:
The products of a number with the natural numbers 1, 2, 3, 4, 5, ... are called the
multiples of the number.
For example: 7x1= 7; 7x 2 = 14; 7x 3 = 21; 7x 4 = 28
So, the multiples of 7 are: 7, 14, 21, 28, and so on.
Example 1
Write down the first ten multiples of 5.
Solution: The first ten multiples of 5 are 5, 10 15, 20, 25, 30, 35, 40, 45, 50.
-Every number has an infinite number of multiples.
> Factors, (divisors):
The product of two numbers is a multiple of each. The two numbers are called factors.
Thus, 8 and 5 are factors of 40 because 8 × 5 = 40.
For example:
20 : 4 = 5
So, 4 is a factor of 20 as it divides exactly into 20.
We could also consider than 20:5 = 4
So, 5 is a factor of 20 as it divides exactly into 20.
Example 1.- List all the factors of 42.
Solution: 42 = 1x42 = 2x21= 3x14 = 6x7
So, the factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42.
Example 2.- Is 7 a factor of 15?
Solution:
15 ÷ 7 =
remainder 1
quotient 2
Clearly 7 does not divide exactly into 15, so 7 is not a factor of 15
Note:
When two or more numbers are multiplied, each number is a factor of the product. If a
number is a factor of a second number, it is also a divisor of the second number.
-Every number has a finite number of factors.
-Every number (a), where a>1, would have at least two positive divisors, ‘1’ and ‘a’.
Unit_1_Divisibility_&_Integers
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Departamento de
Matemáticas
U1_Bilingüe
I.E.S. MIGUEL HERNÁNDEZ
Alhama de Murcia
> Relación de divisibilidad:
Dados dos números a y b, tales que la división a:b es exacta decimos:
1. a es un múltiplo de b.
2. b es un divisor de a.
3. a y b están emparentados por la relación de divisibilidad.
> Divisibility test: (Criterios de divisibilidad)
A number is divisible by 2 if it ends in 0 or an even number: 0, 2, 4, 6 and 8.
Example: 178 is a multiple of 2 because it ends in 8.
A number is divisible by 3 if the sum of its digits is a multiple of 3.
Example: 141 is a multiple of 3 because the sum (addition) of its digits is 1+4+1 = 6.
A number is divisible by 5 if its last digit is a zero or a five.
Example: 340 is a multiple of 5 because it ends en 0.
If the number is divisible by both 2 and 3, then the number is divisible by 6.
If the sum of the digits of the number is divisible by 9, then the number is divisible by 9.
If the number ends in 0, then the number is divisible by 10.
If the sum of the digits in the even position minus the sum of the digits in the uneven
position is 0 or divisible by 11.
1.2: Prime and composite numbers
> Prime number:
If a number has only two different factors, 1 and itself, then the number is said to be a
prime number.
For example, 7 = 1x7
7 is a prime number since it has only two different factors.
¿How many prime numbers can you
find in this chart?
Round it ¡¡¡¡
Note:
A composite number is a whole number greater than 1 with more than two factors
(divisors).
Unit_1_Divisibility_&_Integers
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Departamento de
Matemáticas
U1_Bilingüe
I.E.S. MIGUEL HERNÁNDEZ
Alhama de Murcia
Prime factorization
1.3: Highest common factor
1.4: Least Common Multiple
1.5: Integers
> Definition:
An integer is a whole number (not a fractional number) that can be positive,
negative, or zero.
Examples of integers are: -5, 1, 5, 8, 97
>Positive integers are all the whole numbers greater than zero: 1, 2, 3, 4, 5, ...
>Negative integers are all the opposites of these whole numbers: -1, -2, -3,-4,…
The set of integers, denoted Z, is formally defined as follows:
Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
> The Number Line:
The number line is a line labelled with the integers in increasing order from left
to right, that extends in both directions:
> Important:
For any two different places on the number line, the integer on the right is
greater than the integer on the left.
9 > 4 Is read: nine is ‘greater than’ four,
-7 < 9 Is read: minus seven is ‘less than’ nine.
> Absolute Value of an integer:
The absolute value of an integer is the number without the sign. It is represented
as |a| an it is called absolute value of a.
For example: |+5| = 5 ; |-3| = 3
Geometrically, the absolute value represents the distance from the number to the zero.
> Opposite of an integer:
The opposite of an integer is another integer with the same absolute value but
different sign.
For example: Op (+3) = -3 ; Op (-5) = +5
Unit_1_Divisibility_&_Integers
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Departamento de
Matemáticas
U1_Bilingüe
I.E.S. MIGUEL HERNÁNDEZ
Alhama de Murcia
> Addition and Subtraction of Integers.
To add or subtract two integers we do the following:
If both of them have the same sign, we add their absolute values and we write
their sign.
If they have different sign, we subtract their absolute values and we write by the
result the sign of the integer that has the bigger absolute value.
Rule
Two like signs become a positive sign
Two unlike signs become a negative sign
+(+)
( )
( )
(+)
Example
(+3) + (+2) = 3 + 2 = 5
( ) (-3) = 6 + 3 = 9
( 7) (- ) 7
=5
( ) ( 2) = 8 – 2 = 6
> Multiplication and Division of Integers:
To work out the product, or division, of two integers:
1st. Write + sign if they have like signs or write – sign if they have unlike signs.
2nd. Multiply, or divide, their absolute values.
Remember the rule of the signs when you multiply or divide integers:
· If both numbers have the same sign, the answer is positive -> (+3) · (+5) = +15
(-12) : (-4) = + 3
· If the numbers have different signs, the answer is negative -> (-2) · (+4) = - 8
(+6) : (-6) = - 1
Exercises:
Exercise 1
a) Write down all the multiples of 6 between 20 and 70
b) Write down all the multiples of 7 between 30 and 80
c) Write the three smallest multiples of 8 which are over 50
d) Write the smallest multiple of 37 which is over 500
Exercise 2
Write down all the factors of
a) 60
b) 20
c) 100
Multiples and factors.
http://www.wyzant.com/Help/Math/Elementary_Math/Factors_and_Multiples/Multiples
.aspx
Para aprender la diferencia entre whole numbers and natural numbers.
http://www.mathsisfun.com/whole-numbers.html
Unit_1_Divisibility_&_Integers
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