LEAST COMMON MULTIPLES

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1 Tallahassee Community College 14 LEAST COMMON MULTIPLES Use your math book with this lab. STUDY this lab VERY CAREFULLY! I. Multiples 1. Multiples of 4 are the of 4 and the numbers 1, 2,, 4, 5... (NOTICE 0 x 4 is not considered.) 2. Find the first TEN multiples of 4.. What is the smallest multiple of 4? 4. Explain why there is not a largest multiple of What are the factors of 4? BE SURE YOU KNOW THE MEANINGS OF FACTORS and of MULTIPLES. II. Common Multiples and The Least Common Multiple If Mary likes to dance, sing and play spades and Sarah likes to hike and play spades, we might say that playing spades is a common interest of theirs. Similarly, when two or more numbers have some multiples that are the same numbers, we say these are their common multiples. 1. List the first eight multiples of 6 2. List the first eight multiples of 4. List the numbers that are in both lists 4. These numbers are called the of 6 and What is the smallest of these common multiples of 6 and 4? is called the of 6 and What does LCM stand for? BE SURE YOU KNOW WHAT IS MEANT BY LEAST COMMON MULTIPLE. Listing the multiples of each number and finding the smallest one they have in common helps us to understand what LCM means. It is not always the easiest way to find LCM of two or more numbers. Use the listing method to find the LCM of each set of numbers. 8. Find the LCM of 8, 12, and Find the LCM of 15 and 27.

2 III.Using Prime Factorization to find the Least Common Multiple of numbers. We saw that the LCM of 4 and 6 is 12. Study these prime factorizations. 4 = = 2 NOTICE 12 has 2 2, the factors of = 2 2 It also has 2, the factors of 6. It is not 2 2 2, or all of those factors. ( = 24, a common multiple of 6 and 4, but not the smallest one) Let's find the LCM of 15 and 27 using prime factorizations. Write the prime factorization of each number. 15 = 5 27 = Since and 5 are the only primes used, the LCM of 15 and 27 will have only prime factors of and 5. *You see one factor of in 15, but three factors of in 27. Your LCM will have the greatest number of factors of used. There must be three factors of. *You see one factor of 5 in 15. There is not a factor of 5 in 27. The greatest number of times 5 is used as a factor is one. There will be one factor of 5 in the LCM. The LCM of 15 and 27 is the product of which is 15 5 Finding the LCM will become very easy for you if you: 1. Know what it means 2. Can find it using prime factorizations. Practice finding LCM! Study these examples before trying the lab problems. NOTICE the LCM is given as a number, the product of the primes we used! EXAMPLE 1: Find the LCM of 8, 12, 20 8 = = = LCM = = 120 NOTICE: 120 is a multiple of each of these numbers!

3 EXAMPLE 2: Find the LCM of 6 and 18 6 = 2 18 = 2 LCM = 2 2 = 18 This is a SPECIAL CASE! Think about it. EXAMPLE : Find the LCM of 9 and 10 9 = This is a SPECIAL CASE! 10 = 2 5 Think about it. LCM = 2 5 = 90 EXAMPLE 4: Find the LCM of 12 and = = 5 LCM = = 60 NOTICE is used one time for each number. It is used only one time in the LCM. III. Find the LCM using prime factorizations: 1. Find the LCM of 28 and Find the LCM of 84 and 72.. Find the LCM of 4, 18 and Find the LCM of 26, 9 and Find the LCM of 9 and Find the LCM of 8 and 25. If you have completed Chapter 2, tell when you will use LCM when working with fractions. If you have just begun Chapter 2, make note of each time you use LCM!

4 ANSWERS: I. 1. products 2. 4, 8, 12, 16, 20, 24, 28, 2, 6, (the number itself) 4. There is not a largest natural number. If someone found = 4000; a larger multiple of 4 would be = We can continue this forever, getting a larger multiple of 4 each time. 5. 1, 2, 4 are the only factors of 4. Don't confuse questions about multiples with questions about factors. II. 1. 6; 12; 18; 24; 0; 6; 42; ; 8; 12; 16; 20; 24; 28; 2. 12; 24 (There would be more if the lists above had been continued) 4. common multiples least common multiple (because it's the smallest of the common multiples) 7. Least Common Multiple 8. 8: 8, 16, 24, 2, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, : 12, 24, 6, 48, 60, 72, 84, 96, 108, : 20, 40, 60, 80, 100, LCM = 120 (The first number in all three lists)! 9. 15: 15, 0, 45, 60, 75, 90, 105, 120, 15 27: 27, 54, 81, 108, 15 LCM = 15 III = = 7 LCM = = = = LCM = = = = 2 9 = LCM = 2 2 = 6

5 = = 1 52 = LCM = = = 54 = 2 LCM = 2 = 54 When the larger number is a multiple of the smaller number, the larger number is their LCM. (REMEMBER 9 is a factor of 54; 54 is a multiple of 9.) 6. 8 = = 5 5 LCM = = 200 When there are NO prime factors in common, the LCM is the product of the numbers themselves. LCM of 8 and 25 is 8 25 = 200. This is only true when the only common factor of the numbers is 1. You will use LCM to find the Least Common Denominators of fractions if you add or subtract them or need to tell which fraction is larger.

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