Multiples and factors

Prime factors

1. are factors of a number that are, themselves, prime numbers. 2. There are many methods to find the prime factors of a number, but one of the most common is to use a prime factor tree. 3. Example 4. Write 40 as a product of its prime factors. - Firstly, find two numbers that will multiply together to give 40. - For example 4 x 10 = 40 would be one way of doing this calculation. - Every integer has a unique prime factorisation, so it doesn't matter which factors are chosen to start the factor tree as you will end up with the same answer. - Neither 4 nor 10 is a prime number, and this question is looking for prime factors, so each number must be broken down again into factor pairs. - Continue breaking down the factors into factor pairs until you are only left with prime numbers. Then circle these prime numbers. - The question has asked for a product of prime factors. Write all of the circled prime numbers (found in the prime factor tree) as a product. - This gives 2 x 2 x 2 x 5. This can be written in index form as 2^3 x 5

Roots

1. are the opposite of powers. 2. As 2 squared is 4, then a square root of 4 must be 2. 3. 2^2 = 4. Reversing this gives √(4) = 2. 4. To find square roots or cube roots, work backwards from square numbers and cube numbers. 5. If you know that 15^2 = 225, then you also know that √(225) = 15. 6. If you know that 5^3 = 125, then you also know that 3^√(125) = 5.

Factor pairs

1. are two integers which multiply together to make a particular number. 2. For example, the factor pairs of 12 are 1 and 12; 2 and 6; 3 and 4. 3. This means that the factors of 12 (in order) are 1, 2, 3, 4, 6 and 12. 4. Writing factors in pairs helps to avoid missing any out. 5. All factor pairs will have been found when the next integer to consider is already in the list.

Estimating roots

1. Roots can be estimated by finding the roots of numbers that have integer values above and below the number. 2. Example 3. Estimate the value of √(53). - The square numbers above and below 53 are 49 = 7^2 and 64 = 8^2. This means that the value of √(53) is between 7 and 8, and closer to 7 because 53 is closer to 49 than it is to 64. - So an estimate for √(53) is 7.3 (the exact value is 7.280...).

Multiples

1. The multiples of a number are the values in that number's times table. For example, the multiples of 5 are 5, 10, 15, 20, 25 and so on. 2. There are an infinite amount of multiples of any given number.

Using Venn diagrams

1. A Venn diagram shows the relationship between different sets or categories of data. 2. For example, the following list of numbers can be sorted depending on whether the numbers are even or not, and whether or not they are multiples of 3. - 1, 3, 4, 5, 6, 8, 9, 14, 18 - The numbers 1 and 5 are neither even nor multiples of 3, so they are placed outside the rings of the Venn diagram. - The numbers 6 and 18 are multiples of 3 that are also even numbers, so they are placed in the overlap (or intersection) of the two circles. 3. Venn diagrams can be used to calculate the highest common factor (HCF) and lowest common multiple (LCM) of numbers. 4. Example 5. Find the HCF and LCM of 12 and 180. - Break the numbers into the product of prime factors using prime factor trees. - The product of prime factors for 24 is: 2 x 2 x 2 x 3 - The product of prime factors for 180 is: 2 x 2 x 3 x 3 x 5 - Put each prime factor in the correct place in the Venn diagram. - Any common factors should be placed in the intersection of the two circles. - The highest common factor is found by multiplying together the numbers in the intersection of the two circles. - HCF = 2 x 2 x 3 = 12 - The LCM is found by multiplying together the numbers from all three sections of the circles. - LCM = 2 x 2 x 2 x 3 x 3 x 5 = 360 6. The circles of a Venn diagram do not necessarily contain all the outcomes. 7. Drawing a rectangular frame around the circles enables those outcomes to be recorded outside of the circles but inside the rectangle. 8. The universal set symbol shows that the rectangle includes all the possible outcomes.

Highest common factor

1. A common factor is a factor that is shared by two or more numbers. 2. For example, a common factor of 8 and 10 is 2, as 2 is a factor of 8, and 2 is also a factor of 10. 3. The highest common factor (HCF) is found by finding all common factors of two numbers and selecting the largest one. 4. For example, 8 and 12 have common factors of 1, 2 and 4. - The highest common factor is 4.

Lowest common multiple

1. A common multiple is a number that is a shared multiple of two or more numbers. 2. For example, 24 is a common multiple of 8 and 12, as 24 is in the 8 times tables ( 8 x 3 = 24) and 24 is in the 12 times tables ( 12 x 2 = 24). 3. The lowest common multiple (LCM) is found by listing the multiples of each number and circling any common multiples. 4. The lowest one is the lowest common multiple.

Cube numbers

1. A cube number is the answer when an integer is multiplied by itself, then multiplied by itself again. 2. It is called a cube number because it gives the volume of a cube whose side length is an integer. 3. The first cube number is 1 because 1 x 1 x 1 = 1. 4. The second cube number is 8 because 2 x 2 x 2 = 8. 5. The third cube number is 27 because 3 x 3 x 3 = 27, and so on. 6. The first five cube numbers are: 1, 8, 27, 64 and 125. 7. Example 8. What is the tenth cube number? - 10 cubed = 10 x 10 x 10 = 1000 - So 1000 is the tenth cube number.

Factors

1. A factor is an integer that will divide exactly into another number. 2. For example, 8 is a factor of 24 because 8 will divide into 24 exactly 3 times with no remainder.

Prime numbers

1. A prime number is a number with exactly two factors. 2. A prime number is only divisible by 1 and itself. 3. Another way to think of prime numbers is that they are only ever found as answers in their own times tables. 4. 11 is a prime number because the only factors of 11 are 1 and 11 ( 1 x 11 = 11). 5. No other whole numbers can multiply together to make 11. 6. 15 is not a prime number because the factors of 15 are 1, 3, 5 and 15 ( 1 x 15 = 15, 3 x 5 = 15). 7. 15 has more than 2 factors, so it is not a prime number. 8. 1 is not a prime number as it only has one factor - itself. 9. 2 is the only even prime number. Every even number has 2 as a factor, and so will not be a prime number. 10. There are an infinite number of prime numbers. 11. The prime numbers under 30 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. 12. Example 13. Identify a prime number from this list: 42, 43, 44, 45 - 42 and 44 are even, and so cannot be a prime number as they both have 2 as a factor. - 45 = 5 x 9 and so is not a prime number. - 43 is a prime number as the only factors of 43 are 1 and 43.

Square Numbers

1. A square number is the answer when an integer is multiplied by itself. 2. It is called a square number because it gives the area of a square whose side length is an integer. 3. The first square number is 1 because 1 x 1 = 1 4. The second square number is 4 because 2 x 2 = 4 5. The third square number is 9 because 3 x 3 = 9, and so on. 6. The first fifteen square numbers are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196 and 225. 7. Example 8. What is the twentieth square number? - 20 squared = 20 x 20 = 400 - So 400 is the twentieth square number.

Highest common factors and lowest common multiples using prime factors

1. In some questions the highest common factor (HCF) or lowest common multiple (LCM) of two large numbers may need to be found. 2. It would take a long time to write out all the factors and multiples of 24 and 180 and compare the lists and it would be easy to make a mistake. 3. A more efficient method is to use prime factors.

Using prime factors to find the HCF and LCM

1. Numbers can be broken down into prime factors using prime factor trees. 2. When the prime factors of two numbers are known, they can be compared to calculate HCFs and LCMs. 3. This can be a more efficient method than listing the factors and multiples of large numbers. 4. Example 5. Find the HCF and LCM of 24 and 180. - Start by writing 24 and 180 as the product of their prime factors. - The product of prime factors for 24 are: 2 x 2 x 2 x 3 - The product of prime factors for 180 are: 2 x 2 x 3 x 3 x 5 - To find the HCF, find any prime factors that are in common between the products. Each product contains two 2s and one 3, so use these for the HCF. - HCF = 2 x 2 x 3 = 12 - Cross any numbers used so far from the products. - The product of prime factors for 24 are: 2 - The product of prime factors for 180 are: 3 x 5 - To find the LCM, multiply the HCF by all the numbers in the products that have not yet been used. - LCM = 12 x 2 x 3 x 5 = 360

Estimating powers

1. Powers of any number can be estimated by finding the nearest integers above and below the number. 2. Example 3. Estimate the value of 3.7^3. - 3.7 is between 3 and 4. 3^3 = 27 and 4^3 = 64, so the value of 3.7^3 will be between 27 and 64, and closer to 64 than 27 because 3.7 is closer to 4 than 3 - So an estimate for 3.7^3 would be 50 (the actual value is 50.653).

Powers and roots - Higher

1. Powers, or indices, are ways of writing numbers that have been multiplied by themselves: - 2 x 2 can be written as 2^2 (2 squared) - 2 \times 2 \times 2 can be written as 2^3 (2 cubed) - 2 \times 2 \times 2 \times 2 can be written as 2^4 (2 to the power of 4), and so on 2. The small floating digit is known as the power or index number.