1 ESO - UNIT 03 - DIVISIBILITY
- 1. Unit 03 November 1. DIVISIBILITY. One number is Divisible by other if you divide one number by another the result is a whole number, so the remainder is 𝟎𝟎. We have an exact division. You can distribute equally 250 Books in 5 shelves. 250 ÷ 5 = 50 250 is divisible by 5, because 250 ÷ 5 = 50 is an exact division. If we want to distribute 250 books in 4 shelves: 250 ÷ 4 = 62.5 250 is not divisible by 4, because 62.5 is not a whole number. When we have an exact division, the dividend is called Multiple of the divisor and the quotient. And the quotient and divisor are called Divisors of the dividend. 40 ÷ 8 = 5 40 is multiple of 5 and multiple of 8 8 and 5 are divisors of 40 Two or more numbers are Factors of a number if their product is the number. The number is a multiple of each factor. Each factor is a divisor of the number. 5 and 8 are factors of 40 MATH VOCABULARY: Divisible, Factor, Multiple, Divisor. Axel Cotón Gutiérrez Mathematics 1º ESO 3.1
- 2. Unit 03 November 2. MULTIPLES AND DIVISORS OF A NUMBER. The Multiples of a natural number 𝒂𝒂 can be obtained by multiplying 𝒂𝒂 by any other natural number 𝒌𝒌: 𝒂𝒂 ∙ 𝒌𝒌 = 𝒎𝒎 ⇒ 𝒎𝒎 𝒊𝒊𝒊𝒊 𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎 𝒐𝒐𝒐𝒐 𝒂𝒂 Imagine the number 3, we can find out infinite multiples just by multiplying 3 for any natural number. 3 ∙ 1 = 3; 3 ∙ 2 = 6; 3 ∙ 3 = 9; … ; 3 ∙ 100 = 300;… ; 3 ∙ 5,000 = 15,000,… 3, 6, 9,..., 300,..., 15,000,... are multiples of 3 Any natural number 𝒂𝒂 is always multiple of itself and multiple of 1. 𝒂𝒂 ∙ 𝟏𝟏 = 𝒂𝒂 To obtain the Divisors of a natural number 𝒂𝒂 we look for all the possible exact divisions. Look for the divisors of 30 30 ÷ 1 = 30 ⇒ 1 𝑎𝑎𝑎𝑎𝑎𝑎 30 𝑎𝑎𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑜𝑜𝑜𝑜 30 30 ÷ 2 = 15 ⇒ 2 𝑎𝑎𝑎𝑎𝑎𝑎 15 𝑎𝑎𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑜𝑜𝑜𝑜 30 30 ÷ 3 = 10 ⇒ 3 𝑎𝑎𝑎𝑎𝑎𝑎 10 𝑎𝑎𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑜𝑜𝑜𝑜 30 30 ÷ 4 = 7.5 ⇒ 𝑁𝑁𝑁𝑁𝑁𝑁 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑, 𝑡𝑡ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 30 ÷ 5 = 6 ⇒ 5 𝑎𝑎𝑎𝑎𝑎𝑎 6 𝑎𝑎𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑜𝑜𝑜𝑜 30 Any natural number 𝒂𝒂 is always divisor of itself. And 𝟏𝟏 is divisor of any number. 𝒂𝒂 ÷ 𝒂𝒂 = 𝟏𝟏 𝒂𝒂 ÷ 𝟏𝟏 = 𝒂𝒂 Axel Cotón Gutiérrez Mathematics 1º ESO 3.2
- 3. Unit 03 November 3. DIVISIBILITY RULES. You can use simple divisibility tests to find if one number is divisible by another, without having to do too much calculation. Here are some simple hints: Axel Cotón Gutiérrez Mathematics 1º ESO 3.3
- 4. Unit 03 November 4. PRIME NUMBERS AND COMPOSITE NUMBERS. A Prime number is a number with only two factors: itself and .𝟏𝟏 The first five prime numbers are 2, 3, 5, 7 and 11. is not a prime number because it has only one factor. A Composite number𝟏𝟏 has factors in addition to and itself.𝟏𝟏 8 = 4 ∙ 2 ⇒ 8 𝑖𝑖𝑠𝑠 𝑎𝑎 𝐶𝐶𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑛𝑛𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 The numbers and are neither prime nor composite. A Prime factor is a𝟎𝟎 𝟏𝟏 factor of a number which is also prime. 8 = 4 ∙ 2 ⇒ 2 𝑖𝑖𝑠𝑠 𝑎𝑎 𝑝𝑝𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑓𝑓𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑛𝑛𝑛𝑛 4 𝑖𝑖𝑠𝑠 𝑎𝑎 𝑐𝑐𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑓𝑓𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 4.1. SIEVE OF ERATOSTHENES. Axel Cotón Gutiérrez Mathematics 1º ESO 3.4
- 5. Unit 03 November MATH VOCABULARY: Prime Number, Composite Number, Prime Factor, Sieve of Eratosthenes. 5. PRIME FACTOR DECOMPOSITION. Any number greater than can be written as a product of its prime factors.𝟏𝟏 This is called the Prime Factor Decomposition of the number. There is only one prime factor decomposition for any number. Here are two common methods to find the prime factor decomposition of a number. MATH VOCABULARY: Prime Factor Decomposition, Prime Factorization. Axel Cotón Gutiérrez Mathematics 1º ESO 3.5
- 6. Unit 03 November 6. HIGHEST COMMON FACTOR (HCF) AND LEAST COMMON MULTIPLE (LCM). The Highest Common Factor (HCF) of two or more numbers is the largest number that is factor of all of them. The HCF of 12 and 18 is 6. The Least Common Multiple (LCM) of two or more numbers is the smallest number that is multiple of all of them. The LCM of 12 and 18 is 36. Here are three common methods for finding LCM and HCF: 6.1. METHOD I. To find the HCF of two or more numbers: • List all the factors of each number. • Identify the largest factor that is in all of these lists. 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑜𝑜𝑜𝑜 12 = {1, 2, 3,4, 6, 𝟏𝟏𝟏𝟏} 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑜𝑜𝑜𝑜 24 = {1, 2, 3, 4, 8, 𝟏𝟏𝟏𝟏, 24} 𝐻𝐻𝐻𝐻𝐻𝐻 𝑜𝑜𝑜𝑜 12 𝑎𝑎𝑎𝑎𝑎𝑎 24 = 12 To find the LCM of two or more numbers: • List the first few multiples of each number. • Identify the smallest multiple that is in all of those lists. Axel Cotón Gutiérrez Mathematics 1º ESO 3.6
- 7. Unit 03 November 𝑀𝑀𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑜𝑜𝑜𝑜 12 = {12, 𝟐𝟐𝟐𝟐, 36, 48, … , 1,200,. . . } 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑜𝑜𝑜𝑜 24 = {𝟐𝟐𝟐𝟐, 48, …, 96, . . . } 𝐿𝐿𝐿𝐿𝐿𝐿 𝑜𝑜𝑜𝑜 12 𝑎𝑎𝑎𝑎𝑎𝑎 24 = 24 6.2. METHOD II. You can find the HCF and LCM by writing the prime factor decomposition for each number in a Venn diagram. Find the HCF and LCM of 36 and 28. Find the prime factor decomposition of each number: 36 = 22 ∙ 32 𝑎𝑎𝑛𝑛𝑛𝑛 28 = 22 ∙ 7 Write these prime factors in a Venn Diagram. The common factors are in the middle (the ‘intersection’). The HCF of 36 and 28 is the product of the numbers in the intersection: Axel Cotón Gutiérrez Mathematics 1º ESO 3.7
- 8. Unit 03 November 𝐻𝐻𝐻𝐻𝐻𝐻(28,36) = 2 ∙ 2 = 4 The LCM is the product of all the numbers in the diagram: 𝐿𝐿𝐶𝐶𝐶𝐶(28,36) = 22 ∙ 32 ∙ 7 = 252 6.3. METHOD III. To find the HCF of two or more numbers: • Find the prime factor decomposition of each number. • Multiply the common factors with the lowest exponents. 28 = 22 ∙ 7 𝑎𝑎𝑛𝑛𝑛𝑛 126 = 2 ∙ 32 ∙ 7 𝐻𝐻𝐻𝐻𝐻𝐻(28,126) = 2 ∙ 7 = 14 To find the LCM of two or more numbers: • Find the prime factor decomposition of each number. • Multiply the common factors and the non-common factors with the highest exponents. 28 = 22 ∙ 7 𝑎𝑎𝑛𝑛𝑛𝑛 126 = 2 ∙ 32 ∙ 7 𝐿𝐿𝐿𝐿𝐿𝐿(28,126) = 22 ∙ 32 ∙ 7 = 252 MATH VOCABULARY: Highest Common Factor (HCF), Least Common Multiple (LCM), Venn Diagram. Axel Cotón Gutiérrez Mathematics 1º ESO 3.8