Topics
1. Number Sense
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United States of America
Grade 6

1.05 Problem solving with GCF and LCM

Lesson
Practice

Introduction

We have already learned how to find the greatest common factor (GCF) and the least common multiple (LCM) of two or more numbers. We'll now solve real-life problems involving GCF and LCM.

Problem solving with GCF and LCM

There are many applications of GCF and LCM in real-life. Some examples are when we want to group or split things together, to know when an event will repeat, or when two things will happen again at the same time.

But, how can we tell if a word problem requires us to use GCF or LCM to solve?

When we are asked to find the GCF between two or more numbers, we are being asked what is the biggest number that they can all be divided by that leaves no remainder (or leftovers). When we are asked to find the LCM, we are being asked to find the smallest multiple that is shared by two numbers.

We will use GCF for problems that require us to:

  • Split things to smaller groups or sections like in packs, in bags or in boxes

  • Equally share any quantity of items into their largest grouping

  • Arrange something into rows or columns or sets

Some keywords that can help us identify if a problem requires finding GCF are:

  • Biggest, greatest, highest or maximum

  • Dividing, sharing, distributing or cutting into pieces

We will use LCM to solve problems that ask us to:

  • Find when an event will be repeated or happen again

  • Find the number of pieces to collect multiple items in order to have enough

Some keywords that can help us identify if a problem requires finding LCM are:

  • Smallest, least, or minimum

  • Repeated over and over

  • Next

Let's practice with some problems below.

Examples

Example 1

Mario harvested 18 apples and 12 oranges from his orchard. He wants to group the fruits together in baskets to make identical packs to sell in the market.

a

What is the greatest number of baskets he should use?

Worked Solution
Create a strategy

Analyze the problem by understanding what is asked. Look for keywords to help decide whether finding the GCF or LCM is needed to solve the problem.

Apply the idea

The problem asks for the greatest number of baskets Mario should use.

The key word "greatest" gives us a clue that we will find the GCF.

Listing the factors of 18 and 12:

18:\, 1,\,2,\,3,\,6,\,9,\,18

12:\, 1,\,2,\,3,\,4,\,6,\,12

The greatest common factor is 6.

Mario should use 6 baskets.

b

Determine the number of apples and oranges he should place in each basket.

Worked Solution
Create a strategy

Dividing the number by its GCF determines the number of apples and/or oranges in each basket.

Apply the idea
\displaystyle \text{ Number of apples}\displaystyle =\displaystyle 18\div6
\displaystyle =\displaystyle 3
\displaystyle \text{ Number of oranges}\displaystyle =\displaystyle 12\div6
\displaystyle =\displaystyle 2

There should be 3 apples and 2 oranges in each basket.

Example 2

A party store sells balloons in packs of 10 and balloon sticks in packs of 12.

a

What is the least number of balloons and balloon sticks Jenny should buy so that there will be one balloon for each balloon stick?

Worked Solution
Create a strategy

Analyze the problem by understanding what is asked. Look for keywords to help decide whether finding the GCF or LCM is needed to solve the problem.

Apply the idea

We are asked to find the least number of balloons and balloon sticks that Jenny should buy.

The keyword is "least" which gives us a clue that we solve the problem by getting the LCM.

By using the listing method, we can see that the first seven multiples of 10 and 12 are:

10:\, 10,\,20,\,30,\,40,\,50,\,60,\,70

12:\, 12,\,24,\,36,\,48,\,60,\,72,\,84

The least common multiple of 10 and 12 is 60.

Jenny should buy 60 balloons and ballon sticks.

b

How many packs of balloons and balloon sticks should Jenny buy?

Worked Solution
Create a strategy

Divide 60 by the number of pieces in each pack.

Apply the idea
\displaystyle \text{ Number of packs of balloons}\displaystyle =\displaystyle 60\div 10
\displaystyle =\displaystyle 6
\displaystyle \text{ Number of packs of balloon sticks}\displaystyle =\displaystyle 60\div 12
\displaystyle =\displaystyle 5

Jenny should buy 6 packs of balloons and 5 packs of balloon sticks.

Idea summary

We will use GCF for problems that require us to:

  • Split things to smaller groups or sections like in packs, in bags or in boxes

  • Equally share any quantity of items into their largest grouping

  • Arrange something into rows or columns or sets

We will use LCM to solve problems that ask us to:

  • Find when an event will be repeated or happen again

  • Find the number of pieces to collect multiple items in order to have enough

Some keywords that can help us identify if a problem requires finding GCF or LCM are:

GCFLCM
biggestsmallest
greatestleast
highestminimum
dividingrepeated
sharingnext
distributing
cut into

Outcomes

6.NS.B.4

Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor.

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