# Solutions to Questions on Lowest Common Multiple (LCM)

Detailed solutions and explanations to the questions on Lowest Common Multiple are presented.

 A Lowest Common Multiple Calculator (LCM) that may be used to check answers. Answer the following questions Find the lowest common multiple of 5 and 15. Solution The prime factorization of 5 and 15 are: 5 = 5 15 = 3 × 5 The LCM is given by product of all prime number in the prime factorization with the highest power. Hence LCM of 5 and 15 = 5 1 × 3 1 = 15 Find the lowest common multiple of 8, 12 and 18. Solution The prime factorization of 8, 12 and 18 are: 8 = 2 × 2 × 2 = 2 3 12 = 2 × 2 × 3 = 2 2 × 3 18 = 2 × 3 × 3 = 2 × 3 2 The LCM is given by product of all prime number in the prime factorization with the highest power. LCM of 8, 12 and 18 = 2 3 × 3 2 = 72 Find the lowest common multiple of 70 and 90. Solution The prime factorization of 70 and 90 are: 70 = 2 × 5 × 7 = 2 × 5 × 7 90 = 2 × 3 × 3 × 5 = 2 × 3 2 × 5 The LCM is given by product of all prime number in the prime factorization with the highest power. LCM of 70 and 90 = 2 × 5 × 7× 3 2 = 630 What is the lowest common multiple of 180, 216 and 450? The prime factorization of 180, 216 and 450: 180 = 2 × 2 × 3 × 3 × 5 = 2 2 × 3 2 × 5 216 = 2 × 2 × 2 × 3 × 3 × 3 = 2 3 × 3 3 450 = 2 × 3 × 3 × 5 × 5 = 2 × 3 2 × 5 2 The LCM is given by product of all prime number in the prime factorization with the highest power. LCM of 180, 216 and 450= 2 3 × 3 3 × 5 2 = 5400 a) Find the lowest common multiple (LCM) and the greatest common factor (GCF) of 12 and 16 and compare the products LCM(12,16)×GCF(12,16) and 12×16. b) Find the LCM and GCF of 30 and 45 and compare the products LCM(30,45)×GCF(30,45) and 30×45. c) Find the LCM and GCF of 50 and 100 and compare the products LCM(50,100)×GCF(50,100) and 50×100. Solution a) The prime factorization of 12 and 16 are: 12 = 2 × 2 × 3 16 = 2 × 2 × 2 × 2 GCF of 12 and 16 = 4 LCM of 12 and 16 = 48 Product: LCM(12,16)×GCF(12,16) = 48 × 4 = 192 Product of given numbers: 12 × 16 = 192 The prime two products are equal. b) The prime factorization of 30 and 45 are: 30 = 2 × 3 × 5 45 = 3 × 3 × 5 GCF of 30 and 45 = 15 LCM of 30 and 45 = 90 Product: LCM(30,45)×GCF(30,45) = 90 × 15 = 1350 Product of given numbers: 30 × 45 = 1350 The prime two products are equal. c) The prime factorization of 60 and 160 are: 60 = 2 × 2 × 3 × 5 160 = 2 × 2 × 2 × 2 × 2 × 5 GCF of 60 and 160 = 20 LCM of 60 and 160 = 480 Product: LCM(60,160)×GCF(60,160) = 480 × 20 = 9600 Product of given numbers: 60 × 160 = 9600 The prime two products are equal. It is always true that Given two whole numbers M and N and their CGF and LCM, we have the relationship GCF × LCM = M × N