*COMMON MULTIPLES The multiples of 4 and 6 are Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, ....., ....., ..... Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, ....., ....., ..... Common multiples of both 4 and 6 = 12, 24, 36, ....., ....., ..... *Least common Multiple (LCM): Deifiniton: The least common multiple of two or more given numbers is the lowest (or smallest or least) of their common multiples. Note: If one of the two given numbers is a multiple of the other, then the greater number is the LCM of the given numbers. *EXAMPLES: 1.Common multiples of both 4 and 6 are 12, 24, 36, ....., ....., ..... Least of them is 12. That means 12 is the lowest among the common multiples of both 4 and 6. Lowest Common Multiple (LCM) of 4 and 6 is 12 2.Two bells ring together. If the bells ring at every 3 minutes and 4 minutes respectively. After what interval of time will they ring together again? Solution: First bell rings after every 3 minutes. i.e. Firstbell rings at 3 min, 6, 9, 12, 15, 18, 21, 24, ....., ..... (multiples of 3) Second bell rings after every 4 minutes. i.e., Second bell rings at 4 min, 8, 12, 16, 20, 24, ....., ....., (multiples of 4) Both bells ring together after 12 min., 24 min, ....., ...., (common multiples of both 3 and 4) Least of them (LCM) is 12 min. (That means after 12 minutes they ring together again.) *Methods of Finding LCM: 1. Prime Factorization Method The LCM of 36 and 60 can be found by prime factorization method as follows:- Factors of 36 = |2| × |2| × |3| × 3 Factors of 60 = |2| × |2| × |3|× 5 Step-2:Take the common factors of both: 2 × 2 × 3 Step-3:Take the extra factors of both 36 and 60 i.e. 3 and 5. Step-4:LCM is found by the product of all common prime factors of two numbers and extra prime factors of both. Hence, the LCM of 36 and 60 = (2 × 2 × 3) × 3 × 5 = 180 QUIZ-1: Find the LCM of using appropriate method: i. 12 and 15 ii. 15 and 25 iii. 14 and 21 iv. 10 and 11 v. 48, 56 and 72 vi. 26, 14 and 91. vii. 84, 112, 196 viii. 102, 119, 153 ix. 45, 99, 132, 165 x. 5,6,7 2. Division Method To find the LCM of 24 and 90: Step-1:Arrange the given numbers in a row. Step-2:Then divide by a least prime number which divides at least two of the given numbers and carry forward the numbers which are not divisible by that number if any. Step-3:Repeat the process till no numbers have a common factor other than 1. Step-4:LCM is the product of the divisors and the remaining numbers. Thus, the LCM of 24 and 90 is 2 × 3 × 4 × 15 = 360 | 2 |24, 90 ___|_________ 3 |12, 45 ___|_________ |4, 15 Example: Find the LCM of 21, 35 and 42. Solution: 7 | 21, 35, 42 ____|_____________ 3 | 3, 5, 6 ____|_____________ | 1, 5, 2 Thus, the LCM of 21, 35 and 42 is 7 × 3 × 5 × 2 = 210 QUIZ-2:(Fill in the blanks or MCQS) 1.Three measuring tapes are 64 cm., 72 cm. and 96 cm. What is the least length that can be measured by any of the tapes exactly? 2. What is the capacity of the largest vessel which can empty the oil from three vessels containing 32 litres, 24 litres and 48 litres an exact number of times? 3. Prasad and Raju met in the market on 1st of this month. Prasad goes to the market every 3rd day and Raju goes every 4th day. On what day of the month will they meet again? 4.Find the smallest number which when added to 5 is exactly divisible by 12, 14 and 18. **INTERESTING TO KNOW: RELATIONSHIP BETWEEN LCM AND HCF: -->Product of LCM and HCF of the two numbers = Product of the two numbers. EXAMPLE: Consider the numbers 18 and 27. Factors of 18 = 2 × 3 × 3; Factors of 27 = 3 × 3 × 3 LCM of 18 and 27 is 3 × 3 × 3 × 2 = 54 HCF of 18 and 27 is 3 × 3 = 9 LCM × HCF = 54 × 9 = 486 Product of 18 and 27 = 18 × 27 = 486 Q)Find the LCM of 8 and 12 and then find their HCF using the above relation Solution: LCM of 8 and 12 = 2 × 3 × 4 = 24 4 | 8, 12 ___|________ We know, LCM × HCF = product of the two numbers | 2, 3 HCF = Product of the two numbers __________________________ LCM = 8 X 12 ________ = 4 24 Hence, HCF of 8 and 12 = 4 ***TRY HOTS: 1. Find the LCM and HCF of the following numbers? i. 15, 24 ii. 8, 25 iii. 12, 48 Check their relationship. 2. If the LCM of two numbers is 290 and their product is 7250, what will be its HCF? 3. The product of two numbers is 3276. If their HCF is 6, find their LCM? 4. The HCF of two numbers is 6 and their LCM is 36. If one of the numbers is 12, find the other? #TIPS/TRICKS: 1.The smallest number which is exactly divisible by a, b and c is L.C.M of a, b, c. 2.The L.C.M of two or more numbers is greater than or equal to the greatest number of given numbers. 3.The least number which when divided by a, b and c leaves a remainder R in each case. Required number = (L.C.M of a, b, c) + R