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A Lowest Common Multiple Calculator (LCM) that may be used to check answers.

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

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