Greatest Common Factor and Least Common Multiple Section 5.4 Greatest Common Factor and Least Common Multiple Find the greatest common factor by several methods. Find the least common multiple by several methods. Copyright 06, 0, and 008 Pearson Education, Inc. Copyright 06, 0, and 008 Pearson Education, Inc. Greatest Common Factor The greatest common factor (GCF) of a group of natural numbers is the largest number that is a factor of all of the numbers in the group. Example: 0, 4, and : All have and 4 as common factors (but no higher numbers), so 4 is GCF. (Prime Factors Method) Step Write the prime factorization of each number. Step Choose all primes common to all factorizations, with each prime raised to the least exponent that appears. Form the product of all the numbers in Step ; this product is the greatest common factor. Copyright 06, 0, and 008 Pearson Education, Inc. Copyright 06, 0, and 008 Pearson Education, Inc. 4 Example: Greatest Common Factor by Prime Factors Method Find the greatest common factor of 60 and 50. The prime factorizations are below. 60 = 5 50 = 5 The GCF is 5 = 90. Step Write the numbers in a row. Step Divide each of the numbers by a common prime factor. Try, then, and so on. Divide the quotients by a common prime factor. Continue until no prime will divide into all the quotients. Step 4 The product of the primes in steps and is the greatest common factor. Copyright 06, 0, and 008 Pearson Education, Inc. 5 Copyright 06, 0, and 008 Pearson Education, Inc. 6
Find the greatest common factor of, 8, and 0. 8 0 6 9 5 5 Divide by Divide by No common factors Since there are no common factors in the last row, the GCF is = 6. Copyright 06, 0, and 008 Pearson Education, Inc. 7 (Euclidean Algorithm) To find the greatest common factor of two unequal numbers:. Divide the larger by the smaller.. Note the remainder, and divide the previous divisor by this remainder.. Continue the process until a remainder of 0 is obtained. 4. The greatest common factor is the last positive remainder obtained. Copyright 06, 0, and 008 Pearson Education, Inc. 8 common factor of 60 and 68. Step 60 68 0 common factor of 60 and 68. Step Step 60 68 0 60 Copyright 06, 0, and 008 Pearson Education, Inc. 9 Copyright 06, 0, and 008 Pearson Education, Inc. 0 Least Common Multiple common factor of 60 and 68. The GCF is. Step Step 60 68 0 60 4 0 Done The least common multiple (LCM) of a group of natural numbers is the smallest natural number that is a multiple of all of the numbers in the group. Example: 0 and 50. 0 x 50 = 500, but least common multiple is 50. (5 x 0 = x 50). Copyright 06, 0, and 008 Pearson Education, Inc. Copyright 06, 0, and 008 Pearson Education, Inc.
(Prime Factors Method) Step Write the prime factorization of each number. Step Choose all primes belonging to any factorization, with each prime raised to the power indicated by the greatest exponent that it has in any factorization. Form the product of all the numbers in Step ; this product is the least common multiple. Example: Finding the LCM Find the least common multiple of 60 and 50. The prime factorizations are below. 60 = 5 50 = 5 The LCM is 5 = 5400. Copyright 06, 0, and 008 Pearson Education, Inc. Copyright 06, 0, and 008 Pearson Education, Inc. 4 Step Write the numbers in a row. Step Divide each of the numbers by a common prime factor. Try, then, and so on. Divide the quotients by a common prime factor. When no prime will divide all quotients, but a prime will divide some of them, divide where possible and bring any nondivisible quotients down. Continued on next slide (step continued) Continue until no prime will divide any two quotients. Step 4 The product of the prime divisors in steps and as well as all remaining quotients is the least common multiple. [This is not as bad as it sounds.] Copyright 06, 0, and 008 Pearson Education, Inc. 5 Copyright 06, 0, and 008 Pearson Education, Inc. 6 Find the least common multiple of, 8, and 0. 8 0 6 9 5 5 Divide by Divide by No common factors (Formula) The least common multiple of m and n is given by mn LCM =. greatest common factor of m and n The LCM is 5 = 80. Copyright 06, 0, and 008 Pearson Education, Inc. 7 Copyright 06, 0, and 008 Pearson Education, Inc. 8
Example: LCM Formula Find the LCM of 60 and 50. The GCF is 90. [from earlier example] 60 50 6000 LCM = = = 5400 90 90 Available Methods Greatest Common Factor Least Common Multiple Prime factor method Prime factor method Dividing by prime factors Euclidean method Dividing by prime factors Formula using GCF Copyright 06, 0, and 008 Pearson Education, Inc. 9 Example: (Is this GCF or LCM?) Two traffic signals on adjacent streets in a certain town run on a continuous cycle. The signal on First Street has a cycle time (start of red in one direction to next start of red in same direction) of 00 seconds. The signal on Second Street has a cycle time of 0 seconds. The two signals each begin a northbound red phase at :00 p.m. What is the next time that the signals will begin a cycle at the same time? Example: (Is this GCF or LCM?) We want the number of full cycles at both First and Second Street so that the cycles will again begin at the same time. Next time implies we want the smallest necessary number of cycles (multiples) to accomplish this. So it s a Least Common Multiple problem. LCM of 00 and 0: LCM of 00 and 0: 00 0 5 50 65 0 [done] 00 0 5 50 65 0 [done] LCM = x 5 x 0 x = 00 4
00 seconds is the next time the signals will begin a cycle at the same time. 00 / 60 =.666666 minutes Remember On a quiz or exam, you may use any of these methods for finding the Greatest Common Factor or Least Common Multiple! So, next time is ::40. 5