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1. Revision Description Recall basics of factors and multiples. Reflect and Review 1 is neither prime nor composite, since it has only one factor, i.e., itself. 2. Prime Factorisation Description Writing a number as a product of its prime factors is called Prime Factorisation. Reflect and Review Factorisation of 36 is as follows 36 2×⑱ Prime factorisation of a number is unique. Prime factorisation of a number can be done by dividing it with prime numbers by which it is divisible, till we get quotient as a prime number. The number 1 should not be included in the prime factorisation of any number. 2×⑨ 3×3 Prime factorisation of 720 is as follows: 2 2 2 2 3 3 5 720 360 180 90 45 15 5 1 Thus, 720 = 2 × 2 × 2 × 2 × 3 × 3 × 5. Teasers Answers 1) Find the smallest number which is odd and also prime. 2) Find the number which is even and also prime. 3) Find the smallest number which is odd and also composite. Teasers 1) Draw the factor tree of 28. 2) Write the prime factorisation of 2304. 1) 3 2) 2 3) 9 Answers 1) 28 2×⑭ 2 × 7 2) 2 2 2 2 2 2 2 2 3 3 2304 1152 576 288 144 72 36 18 9 3 1 Thus, 2304 = 2 × 2 × 2×2×2×2×2×2 × 3 × 3. 1 3. Common Factors and Divisibility Rules Description Reflect and Review If a number is divisible by another number, then it is divisible by each of the factors of that number. If a number is divisible by two co-prime numbers, then it is divisible by their product also. If two given numbers are divisible by a number, then their sum is also divisible by that number. If two given numbers are divisible by a number, then their difference is also divisible by that number. 54 is divisible by 6. So, it is also divisible by the factors of 6, i.e., 1, 2 and 3. Teasers Answers 1) A number is divisible by 72. By which other numbers is it also divisible? 1) 1, 2, 3, 4, 6, 8, 9, 12, 18, 24 and 36. 2) A number is divisible by 5 and 7. By which other number is it also divisible? 4. Highest Common Factor (HCF) Description Reflect and Review HCF of a set of numbers is the product of the common prime factors of those numbers. To find the HCF of 50, 75 and 25: 2 50 5 25 5 5 1 3 75 5 25 5 5 1 5 25 5 5 1 So, 50 = 2 × 5 × 5 75 = 3 × 5 × 5 25 = 5 × 5 Thus, HCF of 50, 75 and 25 is 5 × 5 = 25. 2 2) 35. Teasers 1) Find the HCF of the following: a) 162, 216, 270, b) 135, 270, 360. 2) Find the greatest number that will divide 125, 157 and 220 leaving remainders 1, 2 and 3 respectively. Answers 1) a) 54 b) 45 2) 31 For easy solving in case of bigger numbers, the HCF can be calculated by dividing the bigger numbers by the common prime factors. To find the HCF of 64, 96 and 144: 2 64, 96, 144 2 32, 48, 72 2 16, 24, 36 2 8, 12, 18 4, 6, 9 Thus, HCF of 64, 96 and 144 = 2 × 2 × 2 × 2 = 16. 5. Least Common Multiple (LCM) Description Reflect and Review By Prime factorisation, LCM is the product of common and noncommon prime factors. Only one factor should be taken in the place of factors common to at least two of the given numbers. To find the LCM of 12, 8 and 15: By Common division method, LCM is calculated by finding the product of all the divisors till the numbers in the rows are all 1. To find the LCM of 48, 72 and 108: 12 = 2 × 2 × 3 8=2×2×2 15 = 3 × 5 Thus, LCM of 12, 8 and 15 = 2 × 2 × 2 × 3 × 5 = 120. Teasers 1) Find the LCM of the following a) 36, 54, 72 b) 24, 30, 42 Answers 1) a) 216 b) 840 2) 10032 2) Find the smallest 5-digit number which is divisible by the numbers 8, 12 and 16. 2 48, 72, 108 2 24, 36, 54 2 12, 18, 27 2 6, 9, 27 3 3, 9, 27 3 1, 3, 9 3 1, 1, 3 1, 1, 1 Thus, LCM of 48, 72 and 108 = 2 × 2 × 2 × 2 × 3 × 3 × 3 = 432. 3 6. Relation between HCF and LCM of Two Numbers Description Reflect and Teasers Review For any two numbers, HCF × LCM = Product of the numbers For two numbers, 80 and 100: HCF = 20 LCM = 400 HCF × LCM = 20 × 400 = 8000. Product of the numbers = 80 × 100 = 8000. Thus, HCF × LCM = Product of the numbers. 7. Real Life Applications Description Reflect and Review The concept of HCF and LCM can be applied in various real life situations. 4 The longest possible square tile that can be fixed on all the sides of a tank of dimensions 200 m, 120 m and 280 m is 40 m since, HCF of 200, 120 and 280 is 40. If the traffic light at three signals on the way to Atul’s office flashes red once in every 24 sec, 18 sec, and 32 sec respectively, then once in every 288 sec (i.e., in every 4 min 48 sec), all the three signals shows the red light together, since LCM of 24, 18 and 32 is 288. Answers 1) The HCF and LCM of 1) 144 two numbers are 18 and 864 respectively. 2) 1; If one of the numbers Co-primes. is 108, find the other number. 2) If the LCM of two numbers is 1860 and their product is also 1860, then find their HCF. Can you guess the type for the two numbers? Teasers Answers 1) Gina is making flower 1) 8 arrangements. She has 16 roses and 72 daisies. 2) 360 cm × If Gina wants to make 60 cm all the arrangements identical and have no flowers left over, what is the greatest number of flower arrangements she can make? 2) Rectangular tiles of dimensions 24 cm × 15 cm, 36 cm × 10 cm and 30 × 20 cm are available in a shop. What is the minimum length of a square hall in which you can place all the three types of tiles separately without breaking them?