The factors of a number are the numbers that divide exactly into it, with no remainder.

Transcription

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 ȧ. 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. Factors of 20: {±1,±2,±4,±5,±10,±20} Factors and Multiples are linked: Prime Numbers: 12 is divisible by 3 12 is a multiple of 3 3 is a factor of 12. If a number has only two different factors, 1 and itself, then the number is said to be a prime number. If a number has more than two factors, it is called a composite number. Remember, we have already studied the Sieve of Erastothenes that gives us the list of the prime numbers. It starts as follows: Test of divisibility: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, Divisible by 2 A number is divisible by 2 if the last digit is 0, 2, 4, 6 or 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 is divisible by 3 because the sum of the digits is 21 ( =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 is divisible by 4 because 16 is divisible by 4. 21

2 Divisible by 5 A number is divisible by 5 if the last digit is either 0 or 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) 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 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: Digits in odd positions: =24 Digits in odd positions: 9+8+9=26 Digits in even positions: =24 Digits in even positions: 1+2+1=4 The difference is 24-24=0 The difference: 26-4=22 So is divisible by 11. So is divisible by 11. There are a simple way of finding the prime factors of a number: = is the prime factorization of the number

3 Highest Common Factor (HCF) or Greatest Common Factor (GCF): Factors that are common to two or more numbers are said to be common factors. 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. 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 factors with the lowest exponents. Find HCF (360,300): = = So, HCF (360,300) = =60. 23

4 Lowest Common Multiple (LCM) or Least Common Multiple (LCM): Multiples that are common to two numbers are said to be common multiples. 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. 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. Find LCM (18,24) = =2 3 3 So, LCM (18,24)= =72. 24

5 Activities. 1. Express these numbers as products of their prime factors: a) 48 b) 82 c) Calculate the highest common factor (HCF) of: a) 9 and 24 b) 12, 15 and 18 c) 96 and Find out the lowest common multiple (LCM) of: a) 9 and 24 b) 15 and 40 25

6 c) 20 and 30 d) 48, 54 and 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

7 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

8 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