Solutions to Questions on Lowest Common Multiple (LCM)

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

Answer the following questions

1. 
   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
   
2. 
   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
   
3. 
   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
   
4. 
   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
   
5. 
   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

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