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Math 0305 Week #3 Notes Section 4.4 The Common Multiples of are multiples that a set of numbers have in common. So, the common multiples of 3 and 4 are 12, 24, 36, 48, ... The Least Common Multiple (abbreviated L.C.M.) is the smallest multiple the numbers have in common. Thus, the L.C.M. of 3 and 4 is 12. The Least Common Denominator (L.C.D.) is the L.C.M. of the denominators of a set of fractions. L.C.M by Prime Factorization To find the L.C.M. of a number, first find the prime factorization of each number. Next, write down the product of each factor that appears in the prime factorizations and then choose the highest power of the factor in the factorizations. This result is our L.C.M. Example Find the L.C.M. of 49, 7, and 21 Solution: Write down the prime factorization of 49, 7, and 21: 49 = 7•7 = 72 7=7 21 = 3•7 Now, write the product of each factor that appears in the prime factorizations and choose the highest power of the factor in the prime factorizations. So, that would be 3•72 = 3•49 = 147. So, the L.C.M. = 147. L.C.M by Using the Multiples of the Largest Number Start listing the multiples of the largest number until we find a number that is divisible by all the other numbers. Example Find the L.C.M. of 33, 110, and 44 Solution: We will write down the multiples of 110: 110, 220, 330, 440, 550, 660 110, 330, 550 are not divisible by 44 and 220, 440 are not divisible by 33, but 660 is divisible by both 33 and 44 so the L.C.M. = 660. Building Fractions When we reduce fractions to lowest terms, we need to divide out the same number in numerator and denominator. Similarly, if we want to go backwards, we need to multiply the numerator and denominator by the same number. This is referred to as building fractions. Section 4.5 Adding and Subtracting Like Fractions (same denominator) if we add or subtract two fractions with the same denominator, we add or subtract only the numerators. The denominator remains the same. 2 9 Example + 5 9 2+5 9 = = 7 9 Adding Unlike Fractions (different denominators € €and€Subtracting € Unlike fractions are fractions that have different denominators. In order to add or subtract fractions with different denominators, we first find the L.C.M of the denominators, which is now called the L.C.D., the Least Common Denominator. After finding the L.C.D., we build each fraction so it has a denominator equal to the L.C.D. Then we proceed as above. Example 9 14 + 1 3 – 5 21 Solution: We begin by finding the L.C.D. of 14, 3, and 21: 14 = 2•7, 3 = 3, and 21 = 3•7, so the L.C.D. = 2•3•7 = 6•7 = 42. € Next, € we € build the fractions so that they have a denominator equal to 42. Since 42 ÷ 14 = 3, times the top and bottom of the first fraction by 3. Since 42 ÷ 3 = 14, multiply the top and bottom of the second fraction by 14. Since 42 ÷ 21 = 2, times the top and bottom of the third fraction by 2. 9 •3 14 • 3 + 1 • 14 3 • 14 – 5 •2 21 • 2 = 27 42 + Now combine the numerators: 27 42 € € + € € 14 42 – € € € 10 42 = € 31 . 42 € € 14 42 – 10 42