Playing with Numbers

This chapter explores how numbers are built from other numbers. A factor of a number divides it exactly, and a multiple of a number is obtained by multiplying it by a counting number, so 4 is a factor of 12 and 12 is a multiple of 4. You will learn that 1 is a factor of every number, that every number is a factor and a multiple of itself, and that numbers with exactly two factors are prime numbers (2, 3, 5, 7, ...), while numbers with more than two factors are composite. You will use the Sieve of Eratosthenes to find primes up to 100 and learn the tests of divisibility for 2, 3, 4, 5, 6, 8, 9, 10 and 11. The chapter then studies common factors and common multiples, co-prime numbers, and prime factorisation by the factor tree and division methods. Finally, you will find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of numbers and use them to solve real problems about equal grouping and events that repeat together.

Detailed Worksheet: Playing with Numbers Section A - Definitions (10 marks) 1. Define factor and multiple with one example each. (2 marks) 2. What are prime and composite numbers? Why is 1 neither prime nor composite? (2 marks) 3. What are co-prime numbers? Are 18 and 35 co-prime? (2 marks) 4. What is prime factorisation? Write the prime factorisation of 60. (2 marks) 5. Define HCF and LCM of two numbers. (2 marks) Section B - Calculations and Applications (15 marks) 6. Using divisibility tests, check whether 9,07,452 is divisible by (i) 2 (ii) 3 (iii) 4 (iv) 6 (v) 9 (vi) 11. Give the reason in each case. (3 marks) 7. Find the prime factorisation of (i) 980 (ii) 1729 (iii) 288 using the division method. (3 marks) 8. Find the HCF of (i) 18 and 48 (ii) 30 and 42 (iii) 70, 105 and 175. (3 marks) 9. Find the LCM of (i) 12 and 18 (ii) 24 and 90 (iii) 9, 12 and 15. (3 marks) 10. Two tankers contain 850 litres and 680 litres of kerosene. Find the maximum capacity of a container that can measure the kerosene of both tankers an exact number of times. (3 marks) Section C - Diagrams (10 marks) 11. Draw factor trees for 72 and 90 in two different ways each. Write the prime factorisation of each and show that the result does not depend on the way the tree is drawn. (4 marks) 12. Draw a 10 x 10 grid of numbers from 1 to 100 and describe the steps of the Sieve of Eratosthenes. List all prime numbers between 1 and 50. (3 marks) 13. Draw a Venn diagram with two circles showing the factors of 24 and the factors of 36. Write the common factors in the overlap and find the HCF. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Three boys step off together from the same spot. Their steps measure 63 cm, 70 cm and 77 cm. (i) What is the minimum distance each should cover so that all can cover the distance in complete steps? (ii) Is this the HCF or LCM? Explain why. (iii) How many steps will each boy take? (5 marks) 15. A student says, "If a number is divisible by 2 and by 4, it must be divisible by 8." (i) Test this with 12. (ii) Is the statement true? (iii) State the correct rule: if a number is divisible by two co-prime numbers, it is divisible by their product. (iv) Use it to explain why divisibility by 2 and 3 implies divisibility by 6. (5 marks) 16. Renu buys two bags of fertiliser weighing 75 kg and 69 kg. (i) Find the maximum value of weight that can measure the fertiliser an exact number of times. (ii) How many times will each bag be measured? (iii) Find the HCF and LCM of 75 and 69 and verify that their product equals 75 x 69. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show prime factorisation for every HCF and LCM problem, and state the divisibility rule you use.