Easy Ways to Find LCM and GCF

Aan Triono included in category Grade 7

2021-07-31 2026-08-08 817 words 4 minutes

Easy ways to find the LCM and GCF using lists of multiples, prime factorization, and repeated division.

Contents

Introduction

Consider this example. Naomi buys chicken noodles every 2 days, Uciha every 3 days, and Sakiro every 5 days. If they all buy noodles together today, after how many days will they meet again on the same day?

This kind of problem can be solved using the idea of the Least Common Multiple (LCM).

Least Common Multiple

The LCM of two or more numbers can be found in several ways:

listing multiples
prime factorization
repeated division

Listing Multiples

To understand the LCM, look at these multiples:

multiples of 2: 2, 4, 6, 8, 10, 12, ...
multiples of 3: 3, 6, 9, 12, 15, ...
multiples of 5: 5, 10, 15, 20, 25, ...

Numbers that appear in more than one list are called common multiples. The smallest of them is the least common multiple.

Examples:

The LCM of 1 and 3 is 3
The LCM of 2 and 5 is 10
The LCM of 4 and 7 is 28

Prime Factorization

Another way to find the LCM is by prime factorization.

Example: find the LCM of 90 and 168.

Finding the LCM with prime factorization

\begin{aligned} 90 &= 2 \times 3^2 \times 5\\ 168 &= 2^3 \times 3 \times 7 \end{aligned}

To find the LCM, include every prime factor that appears and choose the highest power of each one.

\text{LCM} = 2^3 \times 3^2 \times 5 \times 7 = 2520

LCM Calculator

The interactive calculator for this section is available in the original source.

Repeated Division

The LCM can also be found by repeated division. Divide the numbers step by step until all quotients become 1, then multiply all divisors used.

Example: find the LCM of 9, 15, and 42.

Finding the LCM with repeated division

The result is:

3 \times 2 \times 7 \times 5 \times 3 = 630

So the LCM of 9, 15, and 42 is 630.

Common Factors

A number $a$ is called a factor of an integer $b$ if $a$ divides $b$ exactly. In other words:

b = a \times n

for some integer $n$ .

Examples:

factors of 6: 1, 2, 3, 6
factors of 8: 1, 2, 4, 8
factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

The common factors of 6 and 8 are {1, 2}.

Greatest Common Factor

The Greatest Common Factor (GCF) of two or more numbers is the largest positive integer that divides all of them exactly.

The GCF can be found by:

comparing factor lists
using prime factorization
using repeated division

Example 1

Find the GCF of 20 and 12.

Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 12: 1, 2, 3, 4, 6, 12

The largest common factor is 4, so the GCF of 20 and 12 is 4.

Example 2

Find the GCF of 90 and 168.

Finding the GCF with prime factorization

\begin{aligned} 90 &= 2 \times 3^2 \times 5\\ 168 &= 2^3 \times 3 \times 7 \end{aligned}

To find the GCF, multiply the common prime factors using the lowest powers:

\text{GCF} = 2 \times 3 = 6

Example 3

Find the GCF of 24, 48, and 72.

Finding the GCF with repeated division

Multiply the divisors that divide all three numbers:

2 \times 2 \times 3 = 12

So the GCF of 24, 48, and 72 is 12.

GCF Calculator

The interactive calculator for this section is available in the original source.

Solving Everyday Problems with LCM and GCF

1. Motorbike service schedule

Ani has two motorbikes. One must be serviced every 30 days, the other every 25 days. After how many days will both need service on the same day?

\begin{aligned} 25 &= 5^2\\ 30 &= 2 \times 3 \times 5 \end{aligned}

The LCM is:

2 \times 3 \times 5^2 = 150

So both motorbikes will be serviced together every 150 days.

2. Packing bread into bags

Andi has 15 chocolate breads and 20 banana breads. He wants to pack them into bags with the same composition in each bag. How many bags can be made?

Factors of 15: 1, 3, 5, 15
Factors of 20: 1, 2, 4, 5, 10, 20

The GCF of 15 and 20 is 5, so Andi can make 5 identical bags.

3. Patrol schedule

Mr. Amin patrols every 4 days, Mr. Andi every 6 days, and Mr. Budi every 8 days. If they patrol together on August 1, 2021, when will they patrol together for the second time?

\begin{aligned} 4 &= 2^2\\ 6 &= 2 \times 3\\ 8 &= 2^3 \end{aligned}

The LCM of 4, 6, and 8 is:

2^3 \times 3 = 24

So they will meet again 24 days after August 1, 2021, which is August 25, 2021.

Evaluation

Interactive exercises for this section are available in the original source.

Closing

That completes this overview of how to find the LCM and GCF using lists of multiples, prime factorization, and repeated division. Choose the method that feels clearest, then reinforce it with practice.

Reference: Mathematics textbook for Grade 7 junior secondary students, Ministry of Education and Culture of the Republic of Indonesia, 2016.