Least Common MultipleThis lesson will teach you 3 methods for finding the least common multiple(LCM) of two whole numbers. We will start with a definition of the word multiple The multiples of a number are the answers that you get when you multiply that number by the whole numbers Remember that the whole numbers are all numbers from 0 to infinity whole number = {0, 1, 2, 3, 4, 5, 6, 7, 8,........} For instance, you get the multiples of 4 by multiplying 4 by 0, 1, 2, 3, 4, 5,.... I put the dots to show that the sets of whole numbers continues forever. The answer is { 0, 4, 8, 12, 16, 20, .....} In the same way, the multiples of 9 are all the numbers that you get when you multiply 9 by 0, 1, 2, 3, 4, 5, 6,..... After you do that, you will get {0, 9, 18, 27, 36, 45, 54,....} The LCM of two numbers is the smallest number that is a multiple for both numbers. Method #1: Set intersection method: Example: Find LCM of 6 and 9 First list all the multiples of 6 You get {0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60,....} Next,list all the multiples of 9 You get { 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90....} Pull out all the common multiples or find the intersection of the two sets The common multiples are {18, 36, 54,.....} Looking at the list of common multiples immediately above, you can see that the smallest number that is a multiple of both 6 and 9 is 18. Of course, 36 is also a common multiple of 6 and 9. However, it is not the smallest common multiple. Example: Find LCM of 2 and 3 Multiples of 2 are {0, 2, 4, 6, 8, 10,....} Multiples of 3 are {0, 3, 6, 9, 12, 15....} The least common multiple of 2 and 3 is 6 You can also write LCM(2,3) = 6 Method #2: My teacher's method: find LCM( 6, 9) and LCM (120, 180) Technique: Start by dividing each number by 2. (If 2 does not work, start with 3 instead, and so forth) Keep dividing by 2 until 2 does not work anymore When 2 does not work anymore, divide by 3 When 3 does not work anymore, divide by 4 Keep doing this until you can no longer divide LCM = The product of all the numbers on the left of the red line Method #3: Prime factorisation method Find LCM(120, 180) First, find the prime factorization of the numbers 120 = 12 × 10 = 2 × 2 × 3 × 2 × 5 = 2^{3} × 3^{1} × 5^{1} 180 = 18 × 10 = 2 × 3 × 3 × 2 × 5 = 2^{2} × 3^{2} × 5^{1} The least common multiple will be 2^{x} × 3^{y} × 5^{z} x is the bigger exponent of 2^{3} and 2^{2} y is the bigger exponent of 3^{1} and 3^{2} y is the bigger exponent of 5^{1} and 5^{1} The least common multiple is 2^{3} × 3^{2} × 5 = 2 × 2 × 2 × 3 × 3 × 5 = 360





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