1 eso by Rocío Carmona

NUMBERS IN ENGLISH 1. one 2. two 3. three 4. four 5. five 6. six 7. seven 8. eight 9. nine 10. ten 11. eleven 12. twelve 13. thirteen 14. fourteen 15. fifteen

16. sixteen 17. seventeen 18. eighteen 19. nineteen 20. twenty 21. twenty-one 22. twenty-two 23. twenty-three 24. twenty-four 25. twenty-five 26. twenty-six 27. twenty-seven 28. twenty-eight 29. twenty-nine 30. thirty

Para los números del 21 al 99, si la segunda cifra no es cero ponemos las dos palabras separadas por un guión:

31. thirty-one 32. thirty-two 40. forty 50. fifty 60. sixty hundred (or one hundred)

33. thirty-three, etc. 70. seventy 80. eighty

90. ninety

a hundred and one a hundred and two a hundred and three a hundred and four a hundred and five a hundred and six a hundred and seven a hundred and eight a hundred and nine a hundred and ten a hundred and eleven, etc. 200. two hundred 300. three hundred 400. four hundred 500. five hundred, etc.

101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111.

No añadimos “s” a la palabra hundred en 200, 300, etc. Aunque se puede decir frases como

“There are hundreds of people at the football match” 1000. a thousand 1001. a thousand and one 1002. a thousand and two 1003. a thousand and three 1004. a thousand and four, etc. 2000. two thousand 3000. three thousand 4000. four thousand, etc. 1000000. a million or one million 2000000. two million 3000000. three million, etc.

100. a

327=three hundred and twenty-seven 1 327=one thousand three hundred and twenty-seven 51 327=fifty-one thousand three hundred and twenty-seven 251 327=two hundred and fifty-one thousand three hundred and twenty-seven 5 251 327=five million two hundred and fifty-one thousand three hundred and twenty-seven 8=eight 48=forty-eight 248=two hundred and forty-eight 6 248=six thousand two hundred and forty-eight 26 248=twenty-six thousand two hundred and forty-eight 126 248=one hundred and twenty-six thousand two hundred and forty-eight 2 126 248=two million one hundred and twenty-six thousand two hundred and forty-eight 1. Números naturales. Divisibilidad. DIVISIBILITY.

Divisibility in Natural Numbers: In a division of natural numbers, we can find four elements: D=dividend, d=divisor, q=quotient and r=remainder. Dividend (D) divisor (d) remainder (r) quotient (q) A division is exact, if its remainder is zero. In this case, D=d·q is verified. A division isn't exact, if its remainder isn't zero. In this case, D=d·q+r is verified. When the division between two numbers is exact, we say there is a relation of divisibility between them. D

112.

D is divisible by d.

Examples: The division 78:6 is exact: 78 18 0

6 13

The division is exact, because the remainder is zero. D=d·q ⇒78=6⋅13 78 is divisible by 6. There is a relation of divisibility between 78 and 6.

Multiples of a number: A number b is a multiple of another number a if the division b:a is exact. Examples: 36 is a multiple of 4 but it isn't a multiple of 5: 36 4 0 9 The division is exact

36 5 1 7 The division isn't exact

The multiple of a number is the product generated when that number is multiplied by a natural number. The first multiples of a number are obtained by mulyipling the number by each of the natural numbers: 1, 2, 3, 4, … Multiples of a: a·1, a·2, a·3, a·4, a·5, … And we write it like this: a˙={a ·1, a · 2, a· 3, a · 4, a · 5, ...} Every numbers is a multiple of itself and of 1 ( a⋅1=a ). ˙

Example: 5={5,10,15,20,25, 30,35,40 ...}

Calculate the multiples of 4:

Multiples of 4 → 4 =¿ Demostrate that 4 is multiple of itself and of the unit. Factors of a number: A number a is a factor of another number b if the division b:a is exact. Examples: 6 is a factor of 48, but 7 isn't a factor of 48: 48 6 0 8 The division is exact

48 7 6 6 The division isn't exact

If a number can be expresed as a product of two natural numbers, then the natural numbers are factors of the first number. Examples: 6=6⋅1=3⋅2 . So, the factors of 6 are 1, 2, 3 and 6. A factors is any number that will divide into another number exactly (with no part left over): 8 can be divide by 2 (the factor in this example) 4 times. However, in the total number 8 has several factors: 1, 2, 4 and 8. Examples: Factors of 24: 1, 2, 3, 4, 6, 8, 12 and 24 Factors of 27: 1, 3, 9 and 27 Factors of 56: 1, 2, 4, 7, 8, 14, 28 and 54. Factors of 61: 1 and 61. Numbers that are greater than 1 and have only two factors, 1 and itself are called prime numbers.

It's the same to say: 15 is a multiple of 3 = 3 is a factor of 15 = 15 is divisible by 3

Prime and composite numbers: A prime number (or a prime) is a natural number wich has exactly two distint natural numbers divisors: 1 and itself. If a numbers has more than two divisors, it is called composite number. The number 1 is by definition not a prime number.

“Ilustration showing 11 is a prime number while 12 is not”.

1. 2. 3. 4. 5.

A simple ancient algorithm for finding all prime numbers up to a specified natural number, n ( we are going to do it with n=100), is the Sieve of Erastosthenes: Write down the numbers 1, 2, 3, 4, 5, … , n. (Remember n=100 for us). We will eliminate composites by marking them: Inicially all numbers are unmarked. Mark the number 1 as special (it is neither prime nor composite). Cross out all numbers >2 wich are divisible by 2 (every even number). Find the smallest remainder number >2. It is 3. So cross out all numbers >3 wich are divisible by 3. Find the smallest remainder number >3. It is 5. So cross out all numbers >5 wich are divisible by 5. Continue untill you have crossed out all number divisible by

(in our case

n 100 Prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, …

Factoring numbers: Prime factoring is to factor and then continue factoring a number untill you can no longer reduce the factors into constituent factors any further. Any number can be written as a product of prime numbers in a unique way (except for the order).

Examples: To find the prime factoring of 60: 60 is divisible by 2 → 60:2=30 → 60=2·30 30 is divisible by 2 → 30:2=15 → 60=2·2·15 15 isn't divisible by 2, but 15 is divisible by 3 → 15:3=5 → 60=2·2·3·5 5 isn't divisible by 5, but 5 is divisible by 5 → 5:5=1 → 60=2·2·2·3·5·1

Finally you can write 60 as a product of prime factors: 60= 2·2·2·3·5·1. This can be further simplified using exponents to 60=23⋅3⋅5 . It will better for you to start working with the smallest prime number. We usually write this method like this: 60 2 30 2 15 3 5 5 1 60=22⋅3⋅5 Let's see another example: 24 2 12 2 6

← 24 is divisible by 2 ← 12 is divisible by 2

2 ← 6 is divisible by 2 3 3 ← 3 is divisible by 3

1 24=23⋅3 Greatest common factor (GCF): There are different ways to find the GCF of numbers. Look at them and choose the one you prefer!

Method 1 First list all factors of each number, then list the common factors and choose the largest one:

Method 1: Find the GCF of 12 and 18. The factors of 12 are 1, 2, 3, 4, 6 and 12. The factors of 18 are 1, 2, 3, 6, 9 and 18. The common factors of 12 and 18 are 1, 2, 3 and 6. Although the numbers in bold are all common factors of both 12 and 18, 6 is the greatest common factor. We write GCF (12, 18)=6.

Find the Greatest Common Factor (GCF) of the numbers 24 and 36 with this method. The factors of 24 are: The factors of 36 are: The common factors of 24 and 36 are: So the greatest common factors of 24 and 36 is: GCF (24, 36)= Method 2 To find the GCF of a set of numbers, you must factor each of the numbers into primes. Then for each different prime number in all of the factorizations, do the following … 1.

Count the number of times each prime number appears in all the factorizations.

2. 3.

For each prime number, take the lowest of these counts and write the result. The greatest common factor is the product of all the prime numbers written down. Example: GCF (4,6) = 2, because 4= 22 and 6=2·3, so GCF (4,6)=2. If GCF(a,b) =1, it is said that a and b are relative primes (they don't have any common factors except 1. Method 2: Find GCF(72, 90, 120) 1. Determine the prime factorization of each number: 72 36 18 9 3 1

2 2 2 3 3

90 45 15 5 1

72=23⋅32

2 3 3 5

90=2⋅32⋅5

120 60 30 15 5 1

2 2 2 3 5

120=23⋅3⋅5

2. Take the prime numbers that appears in all the factorizations (Remember taking the lowest number of times they appear). Prime numbers selected: 2 and 3. 3. GCF (72, 90, 120) = 2·3 = 6 Least common multiple (LCM): There are different ways to find the LCM of numbers. Look at them and choose the one you prefer!

Method 1 List the multiples of the larger number and stop when you find a multiple of the other number. This is the LCM. Method 1: Find the LCM of 6 and 9. The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54 … The multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81 ... The common multiples of 6 and 9 are: 18, 36, 54, … So, LCM (6, 9) = 18 Find the Least Common Multiple (LCM) of the numbers 10 and 15 with this method. The multiples of 10 are: The multiples of 15 are: The common multiples of 10 and 15 are: So the least common multiple of 10 and 15 is: LCM (10,15)=

Method 2 To find the LCM of a set of numbers, you must factor each number into primes. Then for each different prime number in all of the factorizations, do the following … Count the number of times each number appears in each of the factorizations. For each prime number, take the largest of these counts and write the result. The least common multiple is the product of all the prime numbers written down.

1. 2. 3.

Example: LCM (4,6) = 12, because 4= 22 and 6=2·3, so LCM (4,6)= 22⋅3=12 . Method 2: Find LCM (16, 24, 40) 1. Determine the prime factorization of each number: 16 8 4 2 1

2 2 2 2

24 12 6 3 1

2 2 2 3

24=23⋅3

16= 24

40 20 10 5 1

2 2 2 5

40=23⋅5

2. Take all the prime numbers that appears in all the factorizations (Remember taking the highest number of times they appear). Primes number selected: 24 , 3 and 5. CALCULATIONS ADDITION 2+4=6 SUBTRACTION 8-5=3 MULTIPLICATION 6·5=30 DIVISION 12:3=4

POWERS (6 is the base and 5 is

the index or exponent) ROOTS 16=4

• • • • • • • • • • •

Two and/plus four is/are/equals six Two added to four makes six What's two and four? It's six. Eight minus five is/are/equals three Eight take away five is three Five from eight leaves/is three Six times five is/equals thirty Six fives are thirty Six multiplied by five is/makes thirty (More formal way) 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

SPECIAL POWERS 52 : Five squared

43 : Four cubed • Six raised to fifth • The square root of sixteen is/equals four.

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

1. Write the following numbers in words as in the example: 3 456: Three thousand four hundred and fifty-six 2. Write the following numbers in words as in the example: 3528: Three thousand five hundred and twenty eight 3. Read the following number out loud: 456

4500

34 760 041

90 045 123

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. 6. Write the missing symbols. Then, write the answers in words. 3___7___4 = 14 _________________________________________ 7. Insert brackets to make the following calculations correct: 5+4路8=37

5+4+3路7=54

8. Calculate the following powers mentally and write them in words: 43 54 112 25 53

= 64 Four cubed equals sixty-four = ___________________________________________ = __________________________________________ = ___________________________________________ = ___________________________________________

103 = __________________________________________ 1002 = _________________________________________

Calculate mentally and write in words as in the example:

16 = 4 The square root of sixteen is four 81 = __________________________________________ 121 = _________________________________________ 900 = _________________________________________ 1600 = ________________________________________ 250000 = ______________________________________ 2 1. 2. 3. 4. 5. 6. 7. 11. 1. 2. 3. 4. 5. 6. 7.

The words in all the statements are jumbled up. Rewrite them so that they make sense. 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 EIGHTEEN INTO ____________________________________ FIFTEEN MAKES NINE TO ADDED SIX ______________________________________

In each statements, the words are in the correct order, but the letters of each word have been jumbled up. Rewrite each sentence. 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, … }

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

1. Plot on the number line and after order them from less to great. -2

-5

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

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

1 =−5

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 7 −13 d) −9 − −6 g)

b) −913 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) −10 7−5−6 8 b) 4−9 13−16−2 11−5 c) 14−13 8−20−12 5 d) −2−3−5 4−1 e) −7 10 30−50 15

3. Calculate: (-6) + (5-2) - (-6+1) = 1st Method (Removing first brackets) = - 6 + 5 â&#x20AC;&#x201C; 2 + 6 â&#x20AC;&#x201C; 1 = 2 2nd Method ( Operating first the expressions into brackets) = - 6 + 3 - (-5) = -6 + 3 +5 = 2

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: 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. 1.

( 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

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) −3 · −4 b) −5 · 4 c) 10 · −3 d) −15 : 3 e) 40 : −8 f) −56 : −7 2. Calculate: a) −5 · −2 · −1 b) 2 · −1 · 4 c) −18 : −2 : 3 d) −20 : 2 : −1 e) 9 · −2 · −1 · 2 f) −5 · −2 · 1 · 8 g) −36 : 9 · −2 h) −15 · −3 : −5

Powers of Integers: Powers are products of equal factors: an=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 −3 4= −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: −2 1=−2 −2 2= −2 · −2 = 4 −2 3 = −2 · −2 · −2 =−8 −2 4= −2 · −2 · −2 · −2 = 16 −2 5= −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.

am⋅an=am Examples:

34⋅37=34 7=311

−2 5⋅ −2 4⋅ −2 = −2 5

4 1

= −2 10

Dividing powers:You can divide powers with the same base by subtracting the exponents.

am :an =am−n Examples:

−3 12 : −3 7= −3 12−7= −3 5

25 :2=25−1=24

Power of a power:You can simplify the power of a power by multiplying the exponents. am n=am

⋅n

⋅

Examples: 42 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=an⋅bn Examples: 34⋅54= 3⋅5 4 =154

[ −2 ⋅3]5= −2 5⋅35

Dividing powers with the same exponent:You can divide powers with the same exponent by dividing the bases. a : b n=an :bn Examples: 154 :34= 15:3 4=54

6:3 5=65 :35

2. Express as just one power: a)

[ −2 2 ]3 : −2 4

b) c) d) e)

24 · 23 : 2 5 32 · 35 ·36 : 34 · 35 84 :82 :82 76 3 · 72 3

64 :[ 28 :27 · 3]3

Square root: The square root of a number a is another number b whose squared is a. a=b when b2=a The number a is called radicand, the symbol √ is called radical and b is called the squared root of a. The numbers with an exact square root are called perfect squares. Examples: =1 because 1 4=2 because 9=3 because 16=4 because 25=5 because 36=6 because

12=1

=11 because 112=121 121

2 =4 32=9 42=16 52=25 62 =36

144=12 because 122=144 169=13 because 132=169 196=14 because 142 =196 225=15 because 152=196 256=16 because 162=256

49=7 because 72 =49 64=8 because 82=64 81=9 because 92 =81 100=10 because 102 =100

289=17 because 172=289 324=18 because 182=324 361=19 because 192=361 400=20 because 202=400

But, be careful! If we are working in the set of integers, a number can have two square roots:

Example: 2 36=±6 , because 62 =36 and −6 =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= 37−62=37−36=1 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 a˙ .

Example: Multiples of 2: 2 ={... ,−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, â&#x20AC;Ś 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. 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, â&#x20AC;Ś) 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.

1º ESO

MATEMÁTICAS

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 23 · 32 is the prime factorization of the number 72. 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. 26 Rocío Carmona

1º ESO

MATEMÁTICAS

Example: Find HCF (360,300): 360 2

300

180 2 150 2 90 2 75 3 45 3 25 5 15 3 5 5 5 5 1 1 300=22 · 3· 52 360=23 ·32 · 5 So, HCF (360,300) = 22 · 3· 5=60 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 24 2 27 Rocío Carmona

1º ESO

MATEMÁTICAS

9 3 3 3 1

12 2 6 2 3 3 1 18=2· 32 24=23 · 3 So, LCM (18,24)= 23 · 32=72 .

28 Rocío Carmona