MATH. Finding The Least Common Multiple of a Set of Numbers

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1 5 Module 11 MATH Finding The Least Common Multiple of a Set of Numbers A DepEd-BEAM Distance Learning Program supported by the Australian Agency for International Development

2 To the Learner Hi! Dear learner. You have learned already, how to find the prime factors of a number in the previous learning module. Today you will be dealing with a new lesson about Finding the Least Common Multiple of a set of numbers. Let s Learn This After learning this module, you are expected to have a full understanding on Finding the Least Common Multiple of a set of numbers. Specifically you should be able to : identify the common multiples of a set of numbers. find the least common multiple of a set of numbers. Write the multiples and Least Common Multiple of a set of numbers. Let s Try This A. Write missing multiples of each pair of numbers. Identify the first three common multiples, then encircle the Least Common Multiple. (number 1 is done for you). Number 1.) Ex.: ) ) ) ) 8 12 Multiples 3,6,9,12,15,18 6,12,18,24,30,30,36 4,,12,16,,24, 8,16,24,,,48 6,12,,24,,36,42,48,54 9,,,36,,54 12,,,48,60,72 24,,72,96,... 8,,,32,40,48,,64,72 12,,36,,60,,84 First Three Common Multiples 6, 12,

3 Let s Study This When we multiply a number by 1, 2, 3, 4, and so on, we get multiples of the number. For example: 5: 5, 10, 15 5 x 1 = 5 5 x 2 = 10 5 x 3 = 15 The product of two counting numbers is the multiple of each of the number. Read and analyze the problem: Mr. Chio repairs their house every 6 years while Mr. Ong repairs their house every 12 years. When will they paint their houses at the same time? This problem calls from the Least Common Multiple (LCM) of 6 and 12. A. To find the LCM, list the multiples of the two numbers. 6: 6, 12, 18, 24, 30, 36 12: 12, 24, 36, 48 Common multiples of 6 and 12 are 12, 24, 36 LCM = 12 Other method to find the LCM of two or more numbers. a. Using Prime Factorization 6: 2 x 3 = 2 x 3 12: 2 x 2 x 3 = 2 2 x x 3 choose the largest number from each set Therefore, the LCM of 6 and 12 is 2x2x3 = 12. b. By Continuous Divisions divide by a prime factor common to two numbers multiply the numbers on the left with the numbers at the bottom to obtain the LCM. Hence, the LCM of 6 and 12 is 2x3x1x2 =

4 Here are more examples. B. Find the Least Common Multiple of 18, 30, 45 a. By Prime Factorization 18 : 2 x 3 x 3 = 2 x : 2 x 3 x 5 = 2 x 3 x 5 45 : 3 x 3 x 5 = 3 2 x 5 2 x 3 2 x 5 Thus, the LCM of 18, 30 and 45 is 2 x 3 x 3 x 5 = 90 b. By Continuous Division Divide 18 and 30 by Carry 45 to the next line Divide 3 and 15 by 3. Carry 5 to the next line Stop dividing when any of two of the numbers have no common factors except 1. Thus, the LCM of 18, 30 and 45 is 2 x 3 x 5 = 90. Let s Do This A. Write the missing multiple and find the LCM. 1.) 10: 10,, 30,, 50 2.) 15: 15,,, 60, 75 4: 4, 8,, 16, 6: 6, 12, 18,, _ 3.) 8: 8,,, 32, 4.) 12: 12,,, 48, 60 10: 10, 20,, 40, 18: 18,, 54, 72, _ B. Find the LCM using the Prime Factorization. (Number 5 is done for you). 5.) 10: 2 x 5 15: 3 x 5 do, LCM of 10and 15 is 2x3x5=

5 6.) 18 : 2 x 3 x 3 8 : 2 x 2 x 2 LCM= x x x x = 7.) 15: 3 x 5 8.) 6 : 2 x 3 45: 3 x 5 x 3 8 : 2 x 2 x2 LCM = x x = 12 : 2 x 2 x 3 LCM = x x x = Let s Do More A. Write the missing number and find the LCM using continuous division. (Number 1 is done for you)

6 B. Read and solve. 5.) I am a multiple of 6 and 7; the sum of my two digits is ) I am an odd number and a multiple of 7; the sum of my two digits is 9. Let s Remember This The Least Common Multiple (LCM) of two or more numbers is the smallest number that is divisible by each of the given numbers. To find the LCM of a set of numbers we can use the List of multiples. Prime factorization and Continuous Division. To find the LCM of a set of numbers using continuous division. 1. Arrange the number in a horizontal line and then divide them by a prime factor common to two or more of the numbers. Any number not divisible by the prime factor is brought down to the next row. 2. Repeat step 1 until no two numbers in a new row has a common factor. 3. The product of all the divisors and the numbers in the final row is the LCM. Let s Test Ourselves A. Write the multiples of each pair of numbers. Then find the Least Common Multiple. 1.) 6: 6, 12,,, 2.) 6: 12, 18,,, 8: 8, 16,,, 10: 10, 20, 30,,, 3.) 5: 5, 10, 15,,,, 4.) 12: 12, 24,,, 7: 7, 14, 21,,,, 9: 9, 18, 27,,,

7 5.) 24: 24,, 48: 48,, 12: 12, 24, 36, B. Find the Least Common Multiple of each set of numbers. Use any method. 6.) 10 and 15 7.) 24 and 40 8.) 27 and 45 9.) 10, 15, ) 12, 15, 30 Let s Enrich Ourselves Additional Exercise: A. Find the List Common Multiple of the set of numbers. Using Continuous Division. 1.) 20 and 45 4.) 24 and 60 2.) 18 and 27 5.) 18, 27 and 45 3.) 12 and 36 Let s Consider This If you got a score of 8 above, Congratulations! Proceed to the next module If you got 5-7, answer the next exercise If you got below 5, review the whole module

8 Answer Key Let s Try This 1.) 2.) 6, 12, 18 8, 16, 24 3.) 18, 16, 24 4.) 24, 48, 72 5.) 24, 48, 72 Let s Do This 1.) 10: 20, 40 2.) 15: 30, 45 3.) 8: 16, 24, 40 4: 12, 20 6: 24, 30 10: 30, 50 LCM = 20 LCM = 30 LCM = 40 4.) 12: 24, 36 5.) 2 x 3 x 5 = 30 18: 36, 90 LCM = 36 6.) LCM = 2 x 2 x 2 x 3 x 3 = ) LCM = 3 x 3 x 5 = 45 8.) LCM = 2 x 2 x 2 x 3 = 24 Let s Do More 2.) 5.) ) 63

9 Let s Test Ourselves A. 1.) 6: 18, 24, 30 2.) 6: 24, 30, 36 3.) 5: 20, 25, 30, 35 8: 24, 32, 40 10: 40, 50, 60 7: 28, 35, 42, 49 LCM = 24 LCM = 30 LCM = 35 B. 6.) 30 7.) ) ) ) 60 4.) 12: 36, 48, 60 5.) 24: 48, 72 9: 36, 45, 54 48: 96, 144 LCM = 36 LCM = 48 Let s Enrich Ourselves - Additional Exercise 1.) ) 54 3.) 36 4.) )

Number System. Introduction. Natural Numbers (N) Whole Numbers (W) Integers (Z) Prime Numbers (P) Face Value. Place Value

Number System. Introduction. Natural Numbers (N) Whole Numbers (W) Integers (Z) Prime Numbers (P) Face Value. Place Value 1 Number System Introduction In this chapter, we will study about the number system and number line. We will also learn about the four fundamental operations on whole numbers and their properties. Natural