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) may be used to check your answers.


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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Updated: 20 January 2017 (A Dendane)

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