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**Multiples, factors and primes**

Contents Multiples, factors and primes Multiples and factors A Prime numbers A A Prime factor decomposition HCF and LCM A

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MULTIPLES and FACTORS

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Multiples A multiple of a number is found by multiplying the number by any whole number. What are the first six multiples of 4? To find the first six multiples of 4 multiply 4 by 1, 2, 3, 4, 5 and 6 in turn to get: Discuss the fact that any given number has infinitely many multiples. We can check whether a number is a multiple of another number by using divisibility tests. Links: N3.1 Divisibility and N3.2 HCF and LCM 4, 8, 12, 16, 20 and Any given number has infinitely many multiples.

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**Multiples patterns on a hundred square**

Explore multiple patterns on the hundred grid. For example, colour all multiples of 3 yellow. Ask pupils to describe and justify the pattern.

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**Rectangular arrangements**

Tell pupils that there are six ways to arrange 12 square counters into a rectangle. Drag 12 square counters into the middle of the board and ask a volunteer to arrange it into a rectangle. The six possibilities are: 1 by 12, 2 by 6, 3 by 4, 4 by 3, 6 by 2 and 12 by 1. Establish that there are actually three different rectangles. The second three are rotations of the first three. State that the numbers 1, 2, 3, 4, 6 and 12 are called the factors of 12. They are all of the whole numbers that divide into 12 exactly. Repeat the activity for other numbers, including square numbers. Some pupils may argue that a square is not a rectangle. Stress that a square is a special type of rectangle which happens to have all of its sides the same length. Introduce prime numbers if required. Establish that if there is only one rectangle possible that is the total number of counters across by one counter down, the number is called a prime number.

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**A factor is a whole number that divides exactly into a given number.**

Finding factors A factor is a whole number that divides exactly into a given number. Factors come in pairs. For example, what are the factors of 30? 1 and 30, 2 and 15, 3 and 10, 5 and 6. Discuss the definition of the word factor. Remind pupils that factors always go in pairs (in the example of rectangular arrangements these are given by the length and the width of the rectangle). The pairs multiply together to give the number. Ask pupils if numbers always have an even number of factors. They may argue that they will because factors can always be written in pairs. Establish, however, that when a number is multiplied by itself the numbers in that factor pair are repeated. That number will therefore have an odd number of factors. Pupils may investigate this individually. Establish that if a number has an odd number of factors it must be a square number. So, in order, the factors of 30 are: 1, 2, 3, 5, 6, 10, 15 and 30.

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Factor finder Complete the ‘factor finder’ by clicking on an empty cell to fill it in. Pupils may make a larger ‘factor finder’ on squared paper. Ask pupils to use the ‘factor finder’ to identify all of the numbers that have two factors. Identify these as prime numbers.

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**Circle and square puzzle**

Explain to pupils that the number in each square is the product of the numbers in the three circles joined to it. Start by finding the number in the middle circle. This number must be a factor of all four numbers in the squares. Using divisibility tests or otherwise find the number in the centre. Discuss with pupils how we can find the numbers in the remaining circle by first dividing by the number in the centre and then using common factors. Clicking on any circle will reveal the solution for that circle. Ensure that during the discussion pupils are using the key words: product, multiple, factor and divisible appropriately.

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PRIME NUMBERS

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Sieve of Eratosthenes Start by defining a prime number as a number that has only two factors: the number one and the number itself. Demonstrate how to find the prime numbers less than 100 using the Sieve of Eratosthenes. Start by selecting the number 1 and colouring it yellow. Explain that this is not a prime number because it only has one factor. We then need to colour all the multiples of 2 not including the number 2 itself. This can be done quickly by selecting all of the multiples of two from the menu at the bottom of the screen and colouring them. 2 is a prime number and should not be coloured. It is therefore necessary to clear the current selection, select the 2 and colour it white. Repeat this process for multiples of 3, not forgetting to colour it white after shading its multiples yellow. Proceed to the next uncoloured (prime number), which is 5. Colour multiples of 5 and then 7, not including 5 and 7 themselves to complete the activity. The next uncoloured number is 11. Ask pupils why we do not need to colour multiples of 11 or any other prime number on the square. Verify that we have found all the prime numbers less than 100 by selecting prime numbers from the menu at the bottom of the screen and showing that they coincide with the uncoloured numbers in the number square.

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Prime numbers If a whole number has two, and only two, factors it is called a prime number. For example, the number 17 has only two factors, 1 and 17. Therefore, 17 is a prime number. The number 1 has only one factor, 1. Therefore, 1 is not a prime number. Establish that 2 is the only even prime number. Ask pupils to name all the prime numbers up to 20. There is only one even prime number. What is it? 2 is the only even prime number.

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**The first 10 prime numbers are:**

2 3 5 7 11 13 17 19 23 29 Remind pupils that 1 is not a prime number because it only has one factor, 1. Pupils should know the first 10 prime numbers.

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**Testing for prime numbers**

Is 107 a prime number? We can check whether or not a number is prime by testing for divisibility by successive numbers. Is 107 divisible by 2? The last digit is a 7 so, no. Is 107 divisible by 3? The digit sum is 8 so, no. We don’t need to check for divisibility by 4 because if 2 doesn’t divide into 107, then no multiple of 2 can divide into it. Ask pupils why we only need to check for divisibility by prime numbers. For example, if a number is not divisible by 2, then it cannot be divisible by any multiple of 2. Is 107 divisible by 5? The last digit is a 7 so, no. We don’t need to check for divisibility by 6 because if 2 doesn’t divide into 107, then no multiple of 2 can divide into it.

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**Testing for prime numbers**

Is 107 a prime number? We can check whether or not a number is prime by testing for divisibility by successive numbers. Is 107 divisible by 7? Dividing by 7 leaves a remainder so no. We don’t need to check for divisibility by 8 because if 2 doesn’t divide into 107, then no multiple of 2 can divide into it. We don’t need to check for divisibility by 9 because if 3 doesn’t divide into 107, then no multiple of 3 can divide into it. We don’t need to check for divisibility by 10 because if 2 doesn’t divide into 107, then no multiple of 2 can divide into it.

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**Testing for prime numbers**

Is 107 a prime number? We can check whether or not a number is prime by testing for divisibility by successive prime numbers. Why don’t we need to check for divisibility by 11? We don’t need to check for divisibility by 11 because we have found that no number below 10 divides into 107. That means that any number that multiplied 11 would have to be bigger than 10. Ask pupils to explain why we do not need to check that 11 divides into 107. By checking that 107 is not divisible by 2, 3, 5 and 7 we have established that no number less than 10 divides into 107. Since 107 is less than 10 × 11. We do not need to check for divisibility by 11. Challenge pupils to find all of the prime numbers between 100 and 200. Since, 10 × 11 is bigger than 107 we can stop here. 107 is a prime number.

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**Testing for prime numbers**

When we are testing whether or not a number is prime, we only have to test for divisibility by prime numbers. We don’t need to check for divisibility by any number bigger than the square root of the number. A number is prime if no prime number less than the square root of the number divides into it. If necessary, remind pupils of the meaning of the square root. Links: N4 Powers and roots – Square roots. Also, all prime numbers greater than 5 must end in a 1, 3, 7 or 9.

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An amazing fact Ask pupils to find the following numbers and write them down on an individual whiteboard. Click on the correct answer on the target board to reveal part of the ‘amazing’ fact. Find: A factor of 18. A multiple of 4. A prime number. A number which is divisible by 15 (remind the class that the number must be divisible by both 3 – the digits will add up to 3 – and 5 – the number will end in a 5 or a 0). A square number (remind pupils what this is if necessary). A triangular number (remind pupils what this is if necessary). A multiple of 14 (so, it must be divisible by both 2 and 7). A factor of both 90 and 120. A cube number (remind pupils what this is if necessary). A factor of 110 (the answer is 22. Remind pupils that we work out 22 × 5 by working out 22 × 10 and halving the answer). A multiple of 18 (remember that it must be divisible by all the factors of 18). The last number remaining is 39. Write down all the factors of 39. The amazing fact assumes that the piece of paper in question is 0.1 mm thick. Folding a piece of paper in half doubles its thickness. If a piece of paper could be folded in half 42 times its thickness would be 242 × 0.1 mm = mm = km (to the nearest km). The average distance from the earth to the moon is about km.

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PRIME FACTORS

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**A prime factor is a factor that is also a prime number.**

Prime factors A prime factor is a factor that is also a prime number. For example, What are the factors of 30? The factors of 30 are: 1 Ask for the factors of 30 before revealing them. Then ask which of these factors are prime numbers. 2 3 5 6 10 15 30 The prime factors of 30 are 2, 3, and 5.

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**Products of prime factors**

2 × 3 × 5 = 30 2 × 2 × 2 × 7 = 56 This can be written as 23 × 7 = 56 3 × 3 × 11 = 99 This can be written as 32 × 11 = 99 First of all, explain verbally that one of the reasons prime numbers are so important is that by multiplying together prime numbers you can make any whole number bigger than one. Go through each of the products. Remind pupils of index notation and how to read it. For example 22 is read as “2 squared” or “2 to the power of 2”. 23 is read as “2 cubed” or “2 to the power of 3” Reveal the fact that every whole number greater than 1 is either a prime number or can be written as a product of two or more prime numbers. This is called the Fundamental Theorem of Arithmetic. Pupils are not expected to know this term, suffice to say it’s important. It is a good reason for defining prime numbers to exclude 1. If 1 were a prime, then the prime factor decomposition would lose its uniqueness. This is because we could multiply by 1 as many times as we like in the decomposition. Link: N4 Powers and roots – powers. Every whole number greater than 1 is either a prime number or can be written as a product of two or more prime numbers.

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**The prime factor decomposition**

When we write a number as a product of prime factors it is called the prime factor decomposition. For example, The prime factor decomposition of 100 is: 100 = 2 × 2 × 5 × 5 Verify that 2 × 2 × 5 × 5 = 100 = 22 × 52 There are 2 methods of finding the prime factor decomposition of a number.

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Factor trees 36 4 9 2 2 3 3 Explain that to write 36 as a product or prime factors we start by writing 36 at the top (of the tree). Next, we need to think of two numbers which multiply together to give 36. Ask pupils to give examples. Explain that it doesn’t matter whether we use 2 × 18, 3 × 12, 4 × 9, or 6 × 6: the end result will be the same. Let’s use 4 × 9 this time. Next, we must find two numbers that multiply together to make 4. Click to reveal two 2s. 2 is a prime number so we draw a circle around it. Now find 2 numbers which multiply together to make 9. Click to reveal two 3s. 3 is a prime number so draw a circle around it. State that when every number at the bottom of each branch is circled we can write down the prime factor decomposition of the number writing the prime numbers in order from smallest to biggest. Ask pupil how we can we write this using index notation (powers) before revealing this. 36 = 2 × 2 × 3 × 3 = 22 × 32

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Factor trees 36 3 12 4 3 2 2 Show this alternative factor tree for 36. The prime factor decomposition is the same. 36 = 2 × 2 × 3 × 3 = 22 × 32

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Factor trees 2100 30 70 6 5 10 7 2 3 2 5 Again, explain that there are many ways to draw the factor tree for 2100 but the final factor decomposition will be the same. The prime factors are written in order of size and then simplified using index notation. 2100 = 2 × 2 × 3 × 5 × 5 × 7 = 22 × 3 × 52 × 7

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Factor trees 780 78 10 2 39 5 2 3 13 Here is another example. 780 = 2 × 2 × 3 × 5 × 13 = 22 × 3 × 5 × 13

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**Dividing by prime numbers**

2 96 2 3 2 48 96 = 2 × 2 × 2 × 2 × 2 × 3 2 24 2 12 = 25 × 3 2 6 Explain verbally that another method to find the prime factor decomposition is to divide repeatedly by prime factors putting the answers in a table as follows: To find the prime factor decomposition of 96 start by writing 96. Click to reveal 96. Now, what is the lowest prime number that divides into 96? Establish that this is 2. Remind pupils of tests for divisibility if necessary. Any number ending in 0, 2, 4, 6, or 8 is divisible by 2. Write the 2 to the left of the 96 and then divide 96 by 2. Click to reveal the 2. This may be divided mentally. Discuss strategies such as halving 90 to get 45 and halving 6 to get 3 and adding 45 and 3 together to get 48. We write this under the 96. Now, what is the lowest prime number that divides into 48? Establish that this is 2 again. Continue dividing by the lowest prime number possible until you get to 1. When you get to 1 at the bottom, stop. The prime factor decomposition is found by multiplying together all the numbers in the left hand column. 3 3 1

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**Dividing by prime numbers**

3 5 7 3 315 315 = 3 × 3 × 5 × 7 3 105 5 35 = 32 × 5 × 7 7 7 Talk through this example as before. Remind pupils that to test for divisibility by 3 we must add together the digits and check whether the result is divisible by 3. 1

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**Dividing by prime numbers**

2 702 2 3 13 3 351 702 = 2 × 3 × 3 × 3 × 13 3 117 3 39 = 2 × 33 × 13 13 13 Here is another example. Ask pupils to find the prime factor decomposition of given numbers. 1

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COMMON MULTIPLES HCF LCM

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Common multiples Multiples of 6 Multiples of 8 12 60 6 18 54 66 102… 8 16 24 24 32 40 48 48 56 64 72 72 80 88 96 96 104 … 30 42 78 90 84 36 Start by asking, What is a multiple? Let’s fill in the multiples of 8 on this straight strip. Reveal the first number, 8, and allow pupils to call out each subsequent multiple before revealing it. Now let’s fill in the multiples of 6. Reveal the 6 and allow pupils to call out subsequent multiples as before. What do we call the numbers where the two strips overlap? They are called common multiples. What is the lowest common multiple of 6 and 8? The lowest common multiple is often called the LCM.

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**Multiples on a hundred grid**

Colour all of the multiples of 3 red, all the multiples of 4 blue and all the multiples of 10 yellow. Draw the pupils’ attention to the fact that the colours change when they overlap. Ask pupils to identify all the common multiples of 3 and 4. What do you notice? What is the lowest common multiple of 3 and 4? Ask pupils to identify all the common multiples of 3 and 10. What do you notice? What is the lowest common multiple of 3 and 10? Ask pupils to identify all the common multiples of 4 and 10. What do you notice? What is the lowest common multiple of 4 and 10? Point out that since 4 and 10 share a common factor, 2, all of the common multiples of 4 and 10 are multiples of 20 (not multiples of 40 as many pupils will expect). What is the lowest common factor of 3, 4 and 10?

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**The lowest common multiple**

The lowest common multiple (or LCM) of two numbers is the smallest number that is a multiple of both the numbers. We can find this by writing down the first few multiples for both numbers until we find a number that is in both lists. For example, Multiples of 20 are : 20, 40, 60, 80, 100, 120, . . . You may like to add that if the two numbers have no common factors (except 1) then the lowest common multiple of the two numbers will be the product of the two numbers. For example, 4 and 5 have no common factors and so the lowest common multiple of 4 and 5 is 4 × 5, 20. Pupils could also investigate this themselves later in the lesson. Multiples of 25 are : 25, 50, 75, 100, 125, . . . The LCM of 20 and 25 is 100.

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**The lowest common multiple**

What is the lowest common multiple (LCM) of 8 and 10? The first ten multiples of 8 are: 8 16 24 32 40 48 56 64 72 80 … The first ten multiples of 10 are: 10 Again go through this method of finding the lowest common multiple. 20 30 40 50 60 70 80 90 100 … The lowest common multiple (LCM) of 8 and 10 is 40.

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**The lowest common multiple**

We use the lowest common multiple when adding and subtracting fractions. For example, Add together 4 9 5 12 and The LCM of 9 and 12 is 36. × 4 × 3 Establish verbally by asking for multiples that the lowest common multiple of 9 and 12 is 36. Remind pupils that to add two fractions together they must have the same denominator. The LCM is the lowest number that both 9 and 12 will divide into. Ask pupils how many 9s ‘go into’ 36. Establish that we must multiply 9 by 4 to get 36 before revealing the first arrow. Now, we’ve multiplied the bottom by 4 so we must multiply the top by 4. Remember, if you multiply the top and the bottom of a fraction by the same number, you do not change its value. 16/36 is just another way of writing 4/9. Repeat this explanation as you convert 5/12 to 15/36. 16/36 plus 15/36 equals 31/36. Can this fraction be simplified? Establish that it cannot. Link: N6 – Calculating with fractions – Adding and subtracting fractions. + 4 9 5 12 = 16 15 31 36 + 36 = 36 × 4 × 3

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Common factor diagram Select pupils to come to the board and drag and drop the numbers down the right hand side in such a way that they are in a circle containing a multiple of that number (24, 36 or 40). If the number divides into two of these numbers then it must be placed in the area where the two circles overlap. If the number is a factor of 24, 36 and 40, it must be placed in the region where all three circles overlap. Numbers which do not divide into 24, 36 or 40 must be placed around the outside of the circle. Once the diagram is complete ask questions such as: What are the common factors of 36 and 40? What are the common factors of 24 and 30? What is the highest common factor of 24 and 40?” If we used different numbers would there always be a number in the section where the three circles overlap? Establish that 1 would always be in the central overlapping section because 1 is a factor of every whole number.

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**The highest common factor**

The highest common factor (or HCF) of two numbers is the highest number that is a factor of both numbers. We can find the highest common factor of two numbers by writing down all their factors and finding the largest factor in both lists. For example, Factors of 36 are : 1, 2, 3, 4, 6, 9, 12, 18, 36. Point out that 3 is a common factor of 36 and 45. As is is 1. However, 9 is the highest common factor. Factors of 45 are : 1, 3, 5, 9, 15, 45. The HCF of 36 and 45 is 9.

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**The highest common factor**

What is the highest common factor (HCF) of 24 and 30? The factors of 24 are: 1 2 3 4 6 8 12 24 The factors of 30 are: 1 2 3 5 6 10 15 30 Once the factors have been revealed remind pupils about factor pairs. 24 is equal to 1 × 24, 2 × 12, 3 × 8, and 4 × 6. 30 is equal to 1 × 30, 2 × 15, 3 × 10, and 5 × 6. The highest common factor (HCF) of 24 and 30 is 6.

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**The highest common factor**

We use the highest common factor when cancelling fractions. For example, Cancel the fraction 36 48 The HCF of 36 and 48 is 12, so we need to divide the numerator and the denominator by 12. Talk through the use of the highest common factor to cancel fractions in one step. Link: N5 Using fractions – Equivalent fractions. ÷12 36 48 3 = 4 ÷12

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**Using prime factors to find the HCF and LCM**

We can use the prime factor decomposition to find the HCF and LCM of larger numbers. For example, Find the HCF and the LCM of 60 and 294. 2 60 2 294 2 30 3 147 Recap on the method of dividing by prime numbers introduced in the previous section. 3 15 7 49 5 5 7 7 1 1 60 = 2 × 2 × 3 × 5 294 = 2 × 3 × 7 × 7

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**Using prime factors to find the HCF and LCM**

60 = 2 × 2 × 3 × 5 294 = 2 × 3 × 7 × 7 60 294 2 7 2 3 5 7 We can find the HCF and LCM by using a Venn diagram. We put the prime factors of 60 in the first circle. Any factors that are common to both 60 and 294 go into the overlapping section. Click to demonstrate this. Point out that we can cross out the prime factors that we have included from 294 in the overlapping section to avoid adding then twice. We put the prime factors of 294 in the second circle. The prime factors which are common to both 60 and 294 will be in the section where the two circles overlap. To find the highest common factor of 60 and 294 we need to multiply together the numbers in the overlapping section. The lowest common multiple is found by multiplying together all the prime numbers in the diagram. HCF of 60 and 294 = 2 × 3 = 6 LCM of 60 and 294 = 2 × 5 × 2 × 3 × 7 × 7 = 2940

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**Using prime factors to find the HCF and LCM**

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