Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
UNIT 1: INTEGERS
Remember how to read numbers:
Complete this table:
Number
Cardinal
Ordinal
1
One
First (1st)
2
Two
Second (2nd)
3
Three
Third (3rd)
4
Fourth (4th)
5
Fifth
6
Sixth
7
Seventh
8
Eighth
9
Ninth
10
Tenth
11
Eleven
Eleventh
12
Twelve
Twelfth
13
Thirteen
14
Fourteen
15
Fifteen
16
Sixteen
17
Seventeen
18
Eighteen
19
Nineteen
20
Twenty
21
Twenty-one
Twenty-first
22
23
24
25
26
27
28
29
1
Unit 1: Integers. Mathematics 2nd E.S.O.
30
Thirty
40
Forty
Teacher: Miguel Angel Hernández
Thirtieth
50
60
70
80
90
100
One hundred
1 000
One thousand
Hundredth
100 000
Hundred thousandth
1 000 000 One million
Millionth
AND is use before the last to figures (tens and units) of a number.
103: a (or one) hundred and three
325: three hundred and twenty-five
2315: two thousand three hundred and fifteen
The words hundred, thousand and million can be used in the singular form with “a” or “one”, but
not alone. “A” is more common in an informal style; “one” is used when we were speaking more
precisely.
I want to live for a hundred years
The journey took exactly one hundred light years
I have a thousand euros
“A” is also common in an informal style with measurement-words.
A kilo of oranges costs a pound
Mix one litre of milk with one kilo of flour ...
SINGULAR O PRURAL?
Number are usually written in singular.
Three hundred euros
Several thousand light years
The plural is only used with dozen, hundred, thousand, million, billion, if they are not modified by
another number or expression (for example a few/several).
Hundreds of pounds
Thousands of light years
PHONE NUMBERS
Each figures is said separately (25 __ two five)
The figure '0' is called oh (304 __ three oh four)
Pause after groups of 3 or 4 figures
234 7809 __ two three four, seven eight oh nine
234 5778 __ two three four, five double seven eight (British English) or
two three four, five seven seven eight (American English)
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Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
ZERO, NOUGHT, NIL, LOVE …
The figure 0 is usually called nought in British English and zero in American English
In measurements, 0 is called zero:
Water freezes at zero degrees Celsius
In team games, zero scores are usually called nil in British English, and zero in American English.
In tennis, the word love is used instead of zero (this is derived from French word “l'oeuf”, because
zero can be egg-shaped):
Spain three Germany nil (zero)
Nadal is winning forty-love
1 BILLION
In English, billion usually means a thousand million: 1 000 000 000
Realize that in Spanish billion means 1 000 000 000 000
PLACE VALUE
Every digit in a number represents a different value depending on its position. For example:
In 53, “5” represents fifty units
In 5234, “5” represents five thousand units
This is the place value table we need to write numbers, no matter how big they are:
BILLION HUNDRED TEN
MILLION MILLION
MILLION HUNDRED
TEN
THOUSAND HUNDRED TEN UNIT
THOUNSAND THOUSAND
CALCULATIONS
ADDITION
2+4=6
•
•
•
Two and/plus four is/are/equals six
Two added to four makes six
What's two and four? It's six.
SUBTRACTION
8-5=3
•
•
•
Eight minus five is/are/equals three
Eight take away five is three
Five from eight leaves/is three
MULTIPLICATION
6·5=30
•
•
•
Six times five is/equals thirty
Six fives are thirty
Six multiplied by five is/makes thirty (More formal way)
DIVISION
12:3=4
•
•
Twelve divided by three is/are/equals four
Three into twelve goes four times (for smaller calculations)
•
•
•
Six to the power of five
Six to the fifth power
Six raised to fifth
•
The square root of sixteen is/equals four.
POWERS
6 (6 is the base and 5 is
the index or exponent)
5
ROOTS
 16=4
3
SPECIAL POWERS
52 : Five squared
4 3 : Four cubed
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
ORDER OF OPERATIONS 1. Parenthesis
(PEMDAS)
2. Exponents (Powers, Roots)
3. Multiplications and Divisions
4. Additions and Subtractions
Remember the sentence: Please, Excuse Me Dear Aunt Sally
Activities.
1. Write the following numbers in words as in the example:
3 456: Three thousand four hundred and fifty-six
90 304: _____________________________________________________
765: ________________________________________________________
237: ________________________________________________________
98 053: ______________________________________________________
134 008: _____________________________________________________
45 004: ______________________________________________________
150 003: _____________________________________________________
2. Write the following numbers in words as in the example:
3528: Three thousand five hundred and twenty eight
86 424: ______________________________________________________
987: ________________________________________________________
3 270: _______________________________________________________
30 001: ______________________________________________________
1 487 070: ____________________________________________________
320 569: ______________________________________________________
20 890 300: ____________________________________________________
3. Read the following number out loud:
456
23 455 678
8 112 345 987
4500
12 342 321
6 560 098 100
34 760 041
4 000 000 000
12 001 234 090
4
90 045 123
20 000 000 000
15 023 497 015
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
4. Write the missing words. Then, write the answers in numbers and symbols:
Ten plus three equals thirteen 10+3=13
Twelve minus six equals _______________________________________
Seven times one equals ________________________________________
Twenty-five divided by five equals _______________________________
Eight plus four minus nine equals ________________________________
5. Write the missing numbers. Then, write the answers in words.
3+8 = 11. Three plus eight equals eleven.
3 · ___ = 30 ___________________________________________
___ -5 =13 ____________________________________________
6: ___ +2 = 4 __________________________________________
___ ·6 – 12 =30 ________________________________________
(3· ___ +3) : 10 =3 ______________________________________
6. Write the missing symbols. Then, write the answers in words.
3___7___4 = 14 _________________________________________
9___2___2 = 20 _________________________________________
25___5___4 = 9 _________________________________________
6___3___2 = 16 _________________________________________
16___4___6 = 10 ________________________________________
7___2___4 = 9 __________________________________________
7. Insert brackets to make the following calculations correct:
5+4·8=37
5+4·8=72
6+15:3=11
6+15:3=7
5+4+3·7=54
16+3·2+5=37
24:4+2·7=28
240:5+7-4·3=8
8. Calculate the following powers mentally and write them in words:
4 3 = 64 Four cube equals sixty-four
54 = ___________________________________________
112 = __________________________________________
25 = ___________________________________________
53 = ___________________________________________
103 = __________________________________________
2
100 = _________________________________________
5
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
9. Calculate mentally and write in words as in the example:
 16 = 4 The square root of sixteen is four
 81 = __________________________________________
 121 = _________________________________________
 900 = _________________________________________
 1600 = ________________________________________
 250000 = ______________________________________
10. The words in all the statements are jumbled up. Rewrite them so that they make sense.
1.
2.
3.
4.
5.
6.
7.
FIVE EQUALS PLUS FOUR NINE ____________________________________________
FIFTEEN AND NINE MAKES SIX ____________________________________________
TAKE THREE SEVEN TEN EQUALS _________________________________________
NINE FROM IS ELEVEN TWO _______________________________________________
TWELVE TIMES THREE IS FOUR ____________________________________________
TIMES TWO GOES NINE EISHTEEN INTO ____________________________________
FIFTEEN MAKES NINE TO ADDED SIX ______________________________________
11. In each statements, the words are in the correct order, but the letters of each word have been
jumbled up. Rewrite each sentence.
1.
2.
3.
4.
5.
6.
7.
WOT SLUP THERE SKAME VIEF ____________________________________________
EVENS DAD ENIN SLAQUE ENTEXIS _______________________________________
TIGHE KATE IXS IS WOT __________________________________________________
VIEF FORM VETLEW IS VENES ____________________________________________
WOT MITES THERE SQUEAL IXS ___________________________________________
HERET MISTE OURF SKAME WELVET ______________________________________
ENTEROUF SUNIM NEVLEE SI REETH ______________________________________
Integers:
The first set of number we knew was the set of Natural Numbers (also called whole numbers):
ℕ = {0, 1, 2, 3, 4, 5, 6, 7, …}
There are many situations in which you need to use numbers below zero, one of these is
temperature, others are money that you can deposit (positive) or withdraw (negative) in a bank,
steps that you can take forwards (positive) or backwards (negative).
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, -5, …
Integers allow us to count and order below and above zero. The set of all Integers is represented by
the letter ℤ = { … -4, -3, -2, -1, 0, 1, 2, 3, 4, … }
6
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
The natural numbers are included in the set of Integers. This fact is represented by the symbol
' ' ⊂' ' .
ℕ⊂ℤ is read ℕ is a subset of ℤ .
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:
For any two different places on the number line, the integer on the right is greater than the integer
on the left.
Examples:
4>-1 is read: “four is greater than minus one”
-3<2 is read: “minus three is less than two”
Opposite of an integer:
The opposite of an integer is the same number with the other sign. The distance from a number to
zero is the same as the distance from its opposite to zero.
The opposite of +5 is -5
The opposite of -7 is +7
Absolute value of an integer:
The absolute value of an integer is the number of units is from zero on the number line.
If the number is positive, the absolute value is the same number.
If the number is negative, the absolute value is the opposite.
The absolute value of an integer is always a positive number (or zero). We specify the absolute
value of a number n in between two vertical bars: ∣n∣ .
Examples:
∣3∣=3
∣−5∣=5
∣4∣=4
7
∣−7∣=7
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
1. Plot on the number line and after order them from less to great.
-2
+8
0
-5
3
Write the opposite and the absolute value of all these numbers.
Adding and Subtracting Integers:
Rules for Addition:
When adding integers with the same sign: We add their absolute values, and give the result with
the same sign.
46=10
−3−6=−9
−5−2=−7
When adding integers with the opposite signs: We subtract their absolute values (we subtract the
smaller absolute value from the larger), and give the result with the sign of the integer with the
larger absolute value.
7−9=−2
8−5=3
−61=−5
Rules for Subtraction:
Subtracting an integer is the same as adding the opposite.
We convert the subtracted integer to its opposite, and add the two integers:
The result of subtracting two integers could be positive or negative.
3−7=3−7=−4
−2−−8=−28=6
8
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
You can use a number line to help you to add or subtract integers:
Calculate 4-6:
Start at 4 and subtract 6 (move 6 units to the left):
The answer is -2.
Calculate -3+7:
Start at -3 and add 7 (move 7 units to the right):
The answer is 4.
1. Calculate:
a) −5−2
d) 7−13
g) −9−−6
b) −913
e) −2−10
h) −8−3
c) 15−12
f) 4−−5
i) 12−3
2. Calculate using one of the methods of the example:
Example:
-3 + 8 – 4 + 2 – 5 =
1st Method (Doing the operations in order) = 5 – 4 + 2 – 5 = 1 + 2 – 5 = 3 – 5 = - 2
2nd Method (Grouping positive and negative) = (8+2)+( - 3 – 4 – 5 ) = 10 – 12 = - 2
a)
b)
c)
d)
e)
−107−5−68
4−913−16−211−5
14−138−20−125
−2−3−54−1
−71030−5015
9
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
3. Calculate:
(-6) + (5-2) - (-6+1) =
1st Method (Removing first brackets) = - 6 + 5 – 2 + 6 – 1 = 2
2nd Method ( Operating first the expressions into brackets) = - 6 + 3 - (-5) = -6 + 3 +5 = 2
a)
b)
c)
d)
e)
f)
g)
h)
8−3−25
2−7−5−4−1−6
7−4−923−1
24−1−7−4−9
−1−23−73−4−2−3−5
−5−104 12−9−8−16
2−36−8−−182−9−1
−9−4−5−7−104−6
4. Calculate:
a)
b)
c)
d)
e)
3−5−7−410−2−−2
5−31−−2−1
8−5−4 −2−3
43−4 −2−6−8
−713−6−8
5. Calculate:
a)
b)
c)
d)
e)
f)
g)
6[ 3−25]
4−[7−51]
5−[2−6−9]
3−[2−4−27 ]−5−1
5−[53−2−9]−3−10
1−2[4−6−−3−4]1
1−[2−5−3−9−1]− 2−3
6. Calculate:
a)
b)
c)
d)
e)
f)
7−2[4−3−1]
−4−3−26−5−34
−2−[−4−15]
−4−13−[ 2−4−−17]
−31−4−[1−2−3]
[4−35]−[ 4−35]
10
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
Multiplying and Dividing Integers:
Rules for Multiplication.
To multiply a pair of integers:
– If both numbers have the same sign (positive or negative), their product is the product of
their absolute values (their product is positive).
– If the numbers have opposite signs, their product is the opposite of the product of their
absolute values (their product is negative).
– If a number is 0, the product is 0.
Look at the chart below:
PRODUCT +
-
+
+
-
-
-
+
5· 4=20
5· −4=−20
−5· 4=−20
−5· −4=20
To multiply any numbers of integers:
1. Count the number of negative integers in the product. If this numbers is even, the product is
positive, but if the number is odd, the product is negative.
2. Take the product of their absolute values.
( If any of the integers in the product is 0, the product is 0).
−4·6· −2=48
−2·−3·−5=−30
0 ·−5.7=0
−5· 2·−2·4 =80
−1· 3· −2 ·−10=−60
−1· −2 ·1 ·−3· −5=30
11
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
Rules for Division:
To divide a pair of integers the rules are the same than for the product:
– If both numbers have the same sign (positive or negative), divide the absolute values of the
first integer by the absolute value of the second integer (the result is positive).
– If the number have opposite signs, divide the absolute value of the first integer by the
absolute value of the second integer, and give the result a negative sign.
Look at the chart below:
DIVISION
+
-
+
+
-
-
-
+
12:3=4
12:−3=−4
−12:3=−4
−12:−3=4
1. Calculate:
a)
b)
c)
d)
e)
f)
−3· −4
−5·4
10· −3
−15:3
40:−8
−56:−7
2. Calculate:
a)
b)
c)
d)
e)
f)
g)
h)
−5·−2 ·−1
2·−1 ·4
−18:−2: 3
−20:2 :−1
9 ·−2·−1 ·2
−5·−2 ·1· 8
−36 :9·−2
−15·−3:−5
12
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
Powers of Integers:
Powers are products of equal factors:
a n=a · a ·...... · a , n times
where a is the base and n is the exponent or index.
Examples:
42=4·4=16
43=4·4· 4=64
4
−3 =−3·−3·−3 ·−3=81
3
−3 =−3· −3· −3=−27
Sign of the power of an integer:
If the base is positive, the sign will be always positive.
If the base is negative, the sign will be positive if the exponent is even, and negative if it is
odd.
•
•
Example:
1
−2 =−2
2
−2 =−2 ·−2=4
−23 =−2· −2·−2=−8
−24=−2·−2·−2 ·−2=16
5
−2 =−2· −2 ·−2·−2· −2=−32
−26 =−2· −2·−2·−2 ·−2·−2=64 …
Operations with powers:
Multiplying powers: You can multiply powers with the same base by adding the exponents.
a m⋅a n=a m n
Examples:
−25⋅−24⋅−2=−2541=−210
34⋅3 7=347=311
Dividing powers:You can divide powers with the same base by subtracting the exponents.
a m :a n =a m−n
Examples:
−312 :−37=−312−7=−35
5
2 :2=2
13
5−1
=2
4
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
Power of a power:You can simplify the power of a power by multiplying the exponents.
a m n=a m⋅n
Examples:
4 25=42 ·5=410
[−34 ]2=−34⋅2=−38
Multiplying powers with the same exponent:You can multiply powers with the same exponent by
multiplying the bases.
a⋅bn=a n⋅b n
Examples:
34⋅5 4=3⋅54 =154
5
5
5
[−2⋅3] =−2 ⋅3
Dividing powers with the same exponent:You can divide powers with the same exponent by
dividing the bases.
a : bn=a n :b n
Examples:
154 : 34=15 :34=54
6 :35=6 5 :35
1. Express as just one power:
a)
b)
c)
d)
e)
f)
g)
h)
2 3 · 25 · 26
2
7
3 ·3 · 3
2
−6 ·−65 ·−64
610 : 65
6
2
−7 :−7
2
3
−2 ·−2 · −2
[−33 ]5
2
[−26 ]
2. Express as just one power:
a)
b)
c)
d)
e)
f)
3
[−22 ] :−24
 2 4 · 23  : 25
 32 · 35 ·36  :  34 · 35 
 84 :8 2  :82
3
3
 76  ·  72 
3
64 : [  28 :27  · 3 ]
14
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
Square root:
The square root of a number a is another number b whose squared is a.
 a=b when b 2=a
The number a is called radicand, the symbol √
of a.
is called radical and b is called the squared root
The numbers with an exact square root are called perfect squares.
Examples:
 1=1 because 12=1
 4=2 because 2 2=4
 9=3 because 32=9
 16=4 because 4 2=16
 25=5 because 52=25
 36=6 because 62 =36
 49=7 because 72 =49
 64=8 because 82=64
 81=9 because 92 =81
 100=10 because 102 =100
 121=11
 144=12
 169=13
 196=14
 225=15
 256=16
 289=17
 324=18
 361=19
 400=20
because
because
because
because
because
because
because
because
because
because
2
11 =121
122=144
2
13 =169
2
14 =196
2
15 =196
2
16 =256
2
17 =289
2
18 =324
2
19 =361
20 2=400
But, be careful! If we are working in the set of integers, a number can have two square roots:
Example:
 36=±6 , because 62 =36 and −62=36
 100=±10 , because 102=100 and −102=100
−4 , it does not exist because any squared integer is negative.
−9 , it does not exist.
Integer square root:
If a radicand is not a perfect square, the square root is not exact. In this case, we talk about integer
square root.
The integer square root of a number a is the greater number b whose squared is less than a.
The remainder of the integer square root is the difference between the radicand a and the squared
of the integer root b.
Examples:
 11≈3 Remainder= 11−32 =11−9=2
 29≈5 Remainder= 29−52=29−25=4
 37≈6 Remainder= 37−6 2=37−36=1
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Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
Order of operations:
Do all operations in brackets (or square brackets) first.
Then, do all the exponents (indexes and roots)
Then, do multiplications and divisions in the order they appear.
Finally, do additions and subtractions in the order they appear.
Easy way to remember:
Parenthesis
Exponents
Multiplications
Divisions
This gives you: PEMDAS: “Please Excuse My Dear Aunt Sally”.
1. Calculate:
a) 5· −8−3·−2
b) 15:−5−22:−2
c) −5·−6−4·−3
d) −14:−2−6:−3
e) −50:[−6−4]
f) 3. [−56:−8]
2. Calculate:
a) 15:−3 ·−1
b) −80:[−8 · 2]
c) [−80 :−8]· 2
d) [9 ·−8]:[−3· −4]
e) 2−5·4:−2
f) −5−40 :−5 ·31
16
Additions
Subtractions
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
3. Calculate:
a) 35−828−12−33·−5
b) 17−[2 ·35−24]· 3−6 ·−5
c) −22 −−32−13
d) 42−42 −−42−42
e) 2 3−3242−17
f) [5 2 ·−32 ]:−15
4. Calculate:
a) 3 2−6573−1024
b) 2 ·[−315−201]−2
c) −23 ·−23−21:1−3
d) 3 ·5−82 ·3−12−53 · 2
e) 2−[ 4−3−7]
f) −5−[23−6· 5−3]
5. Calculate:
a)
 36−  4 64−10
b) −5 ·   49− 81 
c) −1−23−24
d) −17−18 49
e)
  25− 9  : [ 2 :−2 ]
f)
 144− 100  ·  9−8
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Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
6. Calculate:
a) [10−6−24]:3
b) 3:−3−12: 63
c) 62 :[−10−4−4]
d) −3· −4−−24 :615 : −3
e) 10 :−2−−7·−34
f) 1[−3 :−52]−35:−7
Activities.
1. Mount Everest is 29 028 feet above sea level. The Dead Sea is 1 312 feet below sea level.
What is the difference of altitude between these two points?
2. The temperature in Chicago was 4º C at two in the afternoon. If the temperature dropped 12º
C at midnight, what is the temperature now?
3. A submarine was situated 21 000 feet below sea level. If it ascends 1 230 feet, what is its
new position?
4. Aristotle was born in 384 B.C. And died 322 B.C. How old was he when he died?
5. A submarine was situated 1 230 feet below sea level. If it descends 125 feet, what is the new
position?
18
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
6. This is the three-day forecast for Yellowknife (Canada) from the 24th of November 2008.
Today
Nov 24
Tue
Nov 25
Wed
Nov 26
Snow
Cloudy
Snow
-6º C
-13º C
-7ºC
-8ºC
-6ºC
-14ºC
What is the difference between the maximum and minimum temperatures each day?
What are the maximum and minimum temperatures during these three days?
7. This is the three day forecast for Birmingham (UK) from the 24th of November 2008.
Today
Nov 24
Tue
Nov 25
Wed
Nov 26
Rain
Sunny
Partly Cloudy
13º C
2º C
14º C
-2º C
14º C
-1º C
What is the difference between the maximum and minimum temperatures each day?
What are the maximum and minimum temperatures during these three days?
8. The Punic wars began in 264 B.C. and ended in 146 B.C. How long did the Punic Wars last?
19
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
9. This a table with the melting and boiling points of some metals
Metal
Melting Points ºC
Boiling Points ºC
Aluminium
660,32
2519
Iron
1538
2861
Gold
1064,18
2856
Mercury
-38,83
656,73
a) Calculate the difference between the melting and boiling point of each metal.
b) How much warmer is the melting point of mercury than the melting point of iron.
10. On the 2nd of January, the temperature dropped from 3º C at two o'clock in the afternoon to
-11º C at 8 a.m. the next day. How many degrees did the temperature fall?
11. A Greek treasure was buried in the year 164 B.C. and found in 1843 A.D. How long was the
treasure hidden?
12. On the 1st of December, the level of the water in a reservoir was 130 cm above its average
level. On the 1st of July it was 110 cm below its average level. How many cm did the water
level drop in this time?
20
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
Divisibility in the set of integers:
The multiples of a number are obtained multiplying the number by each integer. Usually, the set of
multiples of a number a is written ȧ .
Example: Multiples of 2: 2̇={... ,−6,−4,−2, 0,2,4,6,...}
The factors of a number are the numbers that divide exactly into it, with no remainder.
Example: Factors of 20: {±1,±2,±4,±5,±10,±20}
Factors and Multiples are linked:
12 is divisible by 3 ≡ 12 is a multiple of 3 ≡ 3 is a factor of 12.
Prime Numbers:
If a number has only two different factors, 1 and itself, then the number is said to be a prime
number.
Remember, we have already studied the Sieve of Erastothenes that gives us the list of the prime
numbers. It starts as follows:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, …
Test of divisibility:
Divisible by 2
A number is divisible by 2 if the last digit is 0, 2, 4, 6 or 8.
Example: 2 346 is divisible by 2 because the last digit is 6.
Divisible by 3
A number is divisible by 3 if the sum of the digits is divisible by 3.
Example: 23 457 is divisible by 3 because the sum of the digits is 21 (2+3+4+5+7=21), and 21 is
divisible by 3.
Divisible by 4
A number is divisible by 4 if the number formed by the last two digits is either 00 or divisible by 4.
Example: 24 516 is divisible by 4 because 16 is divisible by 4.
21
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
Divisible by 5
A number is divisible by 5 if the last digit is either 0 or 5.
Example: 9 876 345 is divisible by 5 because the last digit is 5.
Divisible by 6
A number is divisible by 6 if it is divisible by 2 (the last digit is 0, 2, 4, 6 or 8) and it is also
divisible by 3 (the sum of the digits is divisible by 3)
Example: 534 is divisible by 6 because is divisible by 2 (the last digit is 4) and it is divisible by 3
(the sum of the digits 5+3+4=12 is divisible by 3)
Divisible by 10
A number is divisible by 10 if the last digit is 0.
Example: 12 345 890 is divisible by 10 because the last digit is 0.
Divisible by 11
To check if a number is divisible by 11, sum the digits in the odd positions counting from the left
(the first, the third, …) and then sum the remainder digits. If the difference between the sums is
either 0 or divisible by 11, then so is the original number.
Examples: 145 879 635
Digits in odd positions: 1+5+7+6+5=24
Digits in even positions: 4+8+9+3=24
The difference is 24-24=0
So 145 879 635 is divisible by 11.
918 291
Digits in odd positions: 9+8+9=26
Digits in even positions: 1+2+1=4
The difference: 26-4=22
So 918 291 is divisible by 11.
There are a simple way of finding the prime factors of a number:
72 2
36 2
18 2
9 3
3 3
1
72=23 · 32
2 3 · 32 is the prime factorization of the number 72.
22
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
Highest Common Factor (HCF) or Greatest Common Factor (GCF):
Factors that are common to two or more numbers are said to be common factors.
Example:
Factors of 12 are: 1, 2, 3, 4, 6, 12.
Factors of 18 are: 1, 2, 3, 6, 9, 18.
So, common factors of 12 and 18 are 1, 2, 3, 6.
The largest common factor of two or more numbers is called the highest common factor (HCF).
In general, there are two methods for finding the Highest common factor of two or more numbers:
Method I (for small numbers):
List the factor of each number, and find the common factors. The largest of them is the highest
common factor.
Example:
Calculate HCF (8,12):
Factors of 8: 1, 2, 4, 8.
Factors of 12: 1, 2, 3, 4, 6, 12.
So, HCF(8,12)=4.
Method II (general):
To find the highest common factor of two or more numbers:
– Find the prime factorization of each number.
– Choose the common factor with the lowest exponents.
Example:
Find HCF (360,300):
360 2
180 2
90 2
45 3
15 3
5 5
1
360=23 ·32 · 5
So, HCF (360,300) =
300
150
75
25
5
1
2
2
3
5
5
300=22 · 3· 52
2 2 · 3· 5=60 .
23
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
Lowest Common Multiple (LCM) or Least Common Multiple (LCM):
Multiples that are common to two numbers are said to be common multiples.
Example:
Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, …
Multiples of 3 are 3, 6, 9, 12, 15, 18, …
So, common multiples of 2 and 3 are 6, 12, 18, …
The smallest common multiple of two or more numbers is called the lowest common multiple
(LCM).
In general, there are two methods for finding the lowest common multiple of two or more numbers:
Method I (for small numbers):
List the multiple of the largest number and stop when you find a multiple of the other number. This
is the LCM.
Example:
Calculate LCM (8,3):
Multiples of 8 are: 8, 16, 24, 32, 40, …
Multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, …
So, LCM (8,3)=24
Method II (general):
To find the lowest common multiple (LCM) of two or more numbers:
– Find the prime factorization of each number.
– Choose the non common factors and the common factors with the highest exponents.
Example:
Find LCM (18,24)
18 2
9 3
3 3
1
24 2
12 2
6 2
3 3
1
2
18=2· 3
24=23 · 3
So, LCM (18,24)= 2 3 · 32=72 .
24
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
Activities.
1. Express these numbers as products of their prime factors:
a) 48
b) 82
c) 144
2. Calculate the highest common factor (HCF) of:
a) 9 and 24
b) 12, 15 and 18
3. Find out the lowest common multiple (LCM) of:
a) 9 and 24
b) 15 and 40
25
c) 96 and 144
Unit 1: Integers. Mathematics 2nd E.S.O.
c) 20 and 30
Teacher: Miguel Angel Hernández
d) 48, 54 and 72
4. Factorise and then calculate the HCF and LCM of these groups of numbers:
a) 168 and 490
b) 12, 100 and 6
c) 14 and 15
d) 1600 and 1200
e) 294, 1050 and 28
f) 14112, 1080 and 1008
26
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
5. Sandra can pack her books in boxes of 5, 6 and 9, without having any book left. She has less
than 100 books. How many books has she got?
6. Bus routes A and B start at seven o'clock in the morning from the same point. If the bus A
passes the starting point every 24 minutes and the bus every 36 minutes, what time after
seven do their departures coincide again?
7. We want to divide a rectangle of 600 cm by 90 cm into equal squares. Find out the length of
the biggest square in cm. Calculate how many squares we get.
8. We want to cut two ropes that are 20 and 30 m long into pieces as big as possible and of the
same length without wasting everything. What will each piece measure?
9. Iberia has a flight from Madrid to Ankara every 8 days, British Airways one every 12 days
and Easy Jet one every 6 days. One day all three have a flight to Ankara. After how many
days will the three flights coincide again?
27
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
10. In a cycling track one of the cyclists goes round the circuit every 54 seconds and the other
every 72 seconds. They leave the starting line together.
a) How long will it take them to meet again at the starting line for the first time?
b) How many laps will each cyclist have done in that time?
11. What will the side of a square floor tile measure knowing that it has been used to pave the
floor of a garage that is 123 dm long and 90 dm wide? (We have used an exact amount of
floor tiles, without cutting any of them).
12. A baker needs to put 250 cakes and 75 biscuits in boxes as big as possible, with the same
units per boxes but without missing both products in the same box. How many units will
each box contain? How many boxes will he need?
13. A group of students can be organized in lines of 5, 4 and 3 students and there are less than
100. How many students are there?
14. On a Christmas tree, there are two strings of lights, red lights flash every 24 seconds and
green lights every 36 seconds. They starts flashing simultaneously when connect the tree.
When they flash together again?
28
Unit 1: Integers. Mathematics 2nd E.S.O.
Teacher: Miguel Angel Hernández
Keywords:
Natural Numbers= Números Naturales
set= conjunto
Integers= Números Enteros
Positive Number= Número Positivo
Negative Number= Número Negativo
to be included= estar incluido (un conjunto en otro)
subset= subconjunto
Number Line= Recta Numérica
opposite=opuesto
absolute value=valor absoluto
sign= signo
addition= adición, suma to add= sumar
subtraction= resta, substracción
to subtract= restar
multiplication= multiplicación to multiply= multiplicar
division= división to divide= dividir
even number= número par
odd number= número impar
power= potencia
base= base
exponent/index= exponente
square root= raiz cuadrada
radicand= radicando
radical= radical (signo)
perfect squares= cuadrados perfectos
integer square root= raiz cuadrada entera
remainder= resto
Divisibility= Divisibilidad
multiple= múltiplo
factor= divisor
to be divisible by= ser divisible por
prime number= número primo
Tests of divisibility= Criterios de divisibilidad
Highest Common Factor (HCF) (UK) / Greatest Common Factor (GCF) (USA)=
Máximo Común Divisor (m.c.d.)
Lowest Common Multiple (LCM) (UK) / Least Common Multiple (LCM) (USA)= Mínimo
Común Multiplo (m.c.m.)
29