RadfordMathematics.com

Online Mathematics Book

Least Common Multiple (LCM) and Highest Common Factor (HCF)

In this section we learn how to find the least common multiple, lcm, and highest common factor, hcf, which is also known as the greatest common divisor, gcd, of two whole numbers.

For each: we explain the method as well as watch a tutorial illustrating how the method is used and work through an exercise.
By the end of this section you should know how to find the LCM, HCF (or GCD) of any pair of whole numbers.

In order to fully understand what we'll cover here, you may wish to be sure to know how to find the multiples and factors of any whole number. If you're not sure of how to find the factors and multiples of a whole number click HERE.

Least Common Multiple (LCM)

Given two integers \(a\) and \(b\), we define their least common multiple: \[LCM(a,b)\] as the smallest integer which is both:
  • a multiple of \(a\), and:

  • a multiple of \(b\).
For example, the least common multiple of \(2\) and \(3\):
  • multiples of \(2\): \(2,4,6,8,10, \dots \)

  • multiples of \(3\): \(3,6,9,12,15, \dots\)
We can see that the smallest multiple \(2\) and \(3\) both have is \(6\). That's the least common multiple. We write: \[LCM(2,3) = 6\]

Tutorial

In the following tutorial we learn a step-by-step method for finding the least common multiple, LCM, of two whole numbers.

Exercise 1

Find the least common multiple, LCM, of each of the following pairs of integers:

  1. \(3\) and \(4\)

  2. \(4\) and \(6\)

  3. \(5\) and \(6\)

  4. \(8\) and \(12\)

  5. \(10\) and \(12\)

  6. \(4\) and \(7\)

Answers Without Working

  1. \(LCM(3,4)=12\)

  2. \(LCM(5,6) = 30\)

  3. \(LCM(4,6)=12\)

  4. \(LCM(8,12) = 24\)

  5. \(LCM(10,12)=60\)

  6. \(LCM(4,7)=28\)

Solution With Working

Highest Common Factor (HCF)

Given two whole numbers, their highest common factor is the largest factor both of the numbers have in common.

To find the highest common factor we can follow these two steps:

  • Step 1: list the factors of each of the two numbers.

  • Step 2: look at the two lists and find the largest number both lists have in common. That's the highest common factor.
For instance, the highest common factor of \(12\) and \(8\) is \(4\). We would write \(HCF(12,8)=4\).

Tutorial

In the following tutorial we learn a two-step method for finding the HCF (a.k.a the GCD) of two whole numbers. Watch it before attempting to work through the exercise below.

Exercise 2

Find the highest common factor (HCF), aka greatest common divisor (GCD), of each of the following pairs of whole numbers:

  1. \(12\) and \(20\)

  2. \(18\) and \(24\)

  3. \(14\) and \(35\)

  4. \(27\) and \(18\)

  5. \(33\) and \(55\)

  6. \(40\) and \(24\)

Solution Without Working

  1. \(HCF(12,20) = 4\)

  2. \(HCF(18,24) = 6\)

  3. \(HCF(14,35) = 7\)

  4. \(HCF(27,18) = 9\)

  5. \(HCF(33,55) = 11\)

  6. \(HCF(40,24) = 8\)

Solution With Working

If you are unsure of how to find the factors (divisors) of a whole number click HERE.

  1. To find the highest common factor of \(12\) and \(20\), \(HCF(12,20)\), we start by listing all of the factors of both \(12\) and \(20\):

    • The factors of \(12\): \(1, 2, 3, 4, 6, 12\).

    • The factors of \(20\): \(1, 2, 4, 5, 10, 20\).

    Looking at these two lists of factors, the largest number they both have in common is \(4\). So the highest common factor of \(12\) and \(20\) is \(4\), that's: \(HCF(12,20) = 4\).

  2. To find the highest common factor of \(18\) and \(24\), \(HCF(18,24)\), we start by listing all of the factors of both \(18\) and \(24\):

    • Tha factors of \(18\): \(1,2,3,6,9,18\).

    • The factors of \(24\): \(1,2,3,4,6,8,12,24\).

    Looking at these two lists of factors, the largest number both lists have in common is \(6\). So the highest common factor of \(18\) and \(24\) is \(6\), that's: \(HCF(18,24) = 6\)

  3. To find the highest common factor of \(14\) and \(35\), \(HCF(14,35)\), we start by listing the factors of both \(14\) and \(35\):

    • The factors of \(14\): \(1,2,7,14\).

    • The factors of \(35\): \(1,5,7,35\).

    Looking at these two lists of factors, the largest number both lists have in common is \(7\). So the highest common factor of \(14\) and \(35\) is \(7\), that's: \(HCF(14,35)=7\).

  4. To find the highest common factor of \(27\) and \(18\), \(HCF(27,18)\), we start by listing all of the factors of both \(27\) and \(18\):

    • The factors of \(27\): \(1,3,9,27\).

    • The factors of \(18\): \(1,2,3,6,9,18\).

    Looking at these two lists of factors, the largest number both lists have in common is \(9\). So the highest common factor of \(27\) and \(18\) is \(9\), that's: \(HCF(27,18) = 9\).

  5. To find the highest common factor of \(33\) and \(55\), \(HCF(33,55)\), we start by misting all of the factors of both \(33\) and \(55\):

    • The factors of \(33\): \(1,3,11,33\).

    • The factors of \(55\): \(1,5,11,55\).

    Looking at these two lists of factors, the largest number both these lists have in common is \(11\). So the highest common factor of \(33\) and \(55\) is \(11\), that's: \(HCF(33,55)=11\).

  6. To find the highest common factor of \(40\) and \(24\), \(HCF(40,24)\), we start by misting all of the factors of both \(40\) and \(24\):

    • The factors of \(40\): \(1,2,4,5,8,10,20,40\).

    • The factors of \(24\): \(1,2,3,4,6,8,12,24\).

    Looking at these two lists of factors, the largest number both these lists have in common is \(8\). So the highest common factor of \(40\) and \(24\) is \(8\), that's: \(HCF(40,24)=8\).