Math Learner PDF - Factors, Multiples, and Square Roots

## 1.1 Factors, multiples and primes In this section you will - write a positive integer as a product of prime factors - use prime factors to find a highest common factor (HCF) and a lowest common multiple (LCM). Any integer bigger than 1: - is a prime number, or - can be written as a product o...

## 1.1 Factors, multiples and primes In this section you will - write a positive integer as a product of prime factors - use prime factors to find a highest common factor (HCF) and a lowest common multiple (LCM). Any integer bigger than 1: - is a prime number, or - can be written as a product of prime numbers. **Example:** 46 = 2 x 23 49 = 7 x 7 47 is prime 50 = 2 x 5 x 5 48 = 2 x 2 x 2 x 2 x 3 You can use a factor tree to write an integer as a product of its prime factors. **This is how to draw a factor tree for 120.** 1. Write 120. 2. Draw branches to two numbers that have a product of 120. Do not use 1 as one of the numbers. Here we have chosen 12 and 10. 120 = 12 x 10. 3. Do the same with 12 and 10. Here 12 = 3 x 4 and 10 = 2 x 5 4. 3, 2 and 5 are prime numbers, so circle them. 5. Draw two more branches from 4. 4 = 2 x 2. Circle the 2s. 6. Now all the end numbers are prime, so stop. 7. 120 is the product of all the end numbers: 120 = 2 x 2 x 2 x 3 x 5 8. You can check that this is correct using a calculator. You can also write the result like this: 120 = 2^3 x 3 x 5 2^3 means 2 x 2 x 2 and the small 3 is an index. Now check that 75 = 3 x 5^2 You can use products of prime factors to find the HCF and LCM of two numbers. ### Key words - factor tree - highest common factor (HCF) - index - integer - lowest common multiple (LCM) - prime factor ## 1.3 Square roots and cube roots In this section you will... - find the squares of positive and negative integers and their corresponding square roots - find the cubes of positive and negative integers and their corresponding cube roots - learn to recognise natural numbers, integers and rational numbers 5^2 = 25 This means that the **square root** of 25 is 5. This can be written as √25 = 5. This is the only answer in the set of **natural numbers**. However (-5)^2 = -5 x -5 = 25 This means that the integer -5 is also a **square root** of 25. Every positive integer has two **square roots**, one positive and one negative. 5 is the positive **square root** of 25 and -5 is the negative **square root**. No negative number has a square root. For example, the integer -25 has no **square root** because the equation x^2 = -25 has no solution. 5^3 = 125 This means that the **cube root** of 125 is 5. This can be written as ∛125 = 5. You might think -5 is also a **cube root** of 125. However (-5)^3 = -5 x -5 x -5 = (-5 x -5) x -5 = 25 x -5 = -125 So ∛-125 = -5 Every number, positive or negative or zero, has only one **cube root**. ### Key words - cube root - natural numbers - rational numbers - square root ### Tip The **natural numbers** are the counting numbers and zero. ## Worked example 1.3 Solve each equation. a. x^2 = 64 b. x^3 = 64 c. x^3 + 64 = 0 ## 12. Work out these divisions: a. 18 ÷ -6 b. -28 ÷ 4 c. 30 ÷ -6 d. -30 ÷ -10 e. 42 ÷ -6 f. -24 ÷ -4 g. 60 ÷ -5 h. -25 ÷ -5 ## 13. Here are three multiplication pyramids: a. 6 5 -1 b. 12 -8 -2 c. -200 -20 -4 ## 14. Copy and complete each pyramid: a. (3x-4) ÷ -2 c. (-3+15) ÷ -4 b. (2 - 20) ÷ -3 d. 24 ÷ (2x - 4) ## 15. Find the value of x: a. x ÷ -4 = 8 c. 16 - x = -2 b. x - 3 = -15 d. -15 - x = 3 ## 16. Round these numbers to the nearest whole number to estimate the answer: a. -8.75 - 2.8 c. -28.2 - -3.8 b. 18.1 - -5.9 d. -35.2 - -6.9 ## 17. Round these numbers to the nearest 10 to estimate the answer: a. -48 x -29 c. -71.4 + -11.8 b. -18.1 x 61.5 d. -99.4 + 19 ## Summary checklist ☐ I can multiply two negative integers. ☐ I can divide any integer by a negative integer.