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
Links and References
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