Greatest Common Factor and Least Common Multiple

Hi, and welcome to this video covering the least common multiple and the greatest common factor!

As you know, there are times when we have to algebraically “adjust” how a number or an equation appears in order to proceed with our math work. We can use the greatest common factor and the least common multiple to do this. The greatest common factor (GCF) is the largest number that is a factor of two or more numbers, and the least common multiple (LCM) is the smallest number that is a multiple of two or more numbers.

To see how these concepts are useful, let’s look at adding fractions. Before we can add fractions, we have to make sure the denominators are the same by creating an equivalent fraction:

\(\frac{2}{3}+\frac{1}{6} \rightarrow \frac{2}{3} \times \frac{2}{2}\)\(+\frac{1}{6} \rightarrow \frac{4}{6} +\frac{1}{6}=\frac{5}{6}\)

In this example, the least common multiple of 3 and 6 must be determined. In other words, “What is the smallest number that both 3 and 6 can divide into evenly?” With a little thought, we realize that 6 is the least common multiple, because 6 divided by 3 is 2 and 6 divided by 6 is 1. The fraction \(\frac{2}{3}\) is then adjusted to the equivalent fraction \(\frac{4}{6}\) by multiplying both the numerator and denominator by 2. Now the two fractions with common denominators can be added for a final value of \(\frac{5}{6}\).

In the context of adding or subtracting fractions, the least common multiple is referred to as the least common denominator.

In general, you need to determine a number larger than or equal to two or more numbers to find their least common multiple.

It is important to note that there is more than one way to determine the least common multiple. One way is to simply list all the multiples of the values in question and select the smallest shared value, as seen here:

Least common multiple of 8, 4, 6

\(8\rightarrow 8,16,24,32,40,48\)
\(4\rightarrow 4,8,12,16,20,24,28,32\)
\(6\rightarrow 6,12,18,24,30,36\)

This illustrates that the least common multiple of 8, 4, and 6 is 24 because it is the smallest number that 8, 4, and 6 can all divide into evenly.

Another common method involves the prime factorization of each value. Remember, a prime number is only divisible by 1 and itself.

Once the prime factors are determined, list the shared factors once, and then multiply them by the other remaining prime factors. The result is the least common multiple:

\(30=2\times 2\times 3\times 3\)
\(90=2\times 3\times 3\times 5\)

\(\text{LCM}=2\times 3\times 3\times 2\times 5\)

The least common multiple can also be found by common (or repeated) division. This method is sometimes considered faster and more efficient than listing multiples and finding prime factors. Here is an example of finding the least common multiple of 3, 6, and 9 using this method:

Divide the numbers by the factors of any of the three numbers. 6 has a factor of 2, so let’s use 2. Nine and 3 cannot be divided by 2, so we’ll just rewrite 9 and 3 here. Repeat this process until all of the numbers are reduced to 1. Then, multiply all of the factors together to get the least common multiple.

2369
3339
3113
111

LCM \(=2\times 3\times 3=18\)

Now that methods for finding least common multiples have been introduced, we’ll need to change our mindset to finding the greatest common factor of two or more numbers. We will be identifying a value smaller than or equal to the numbers being considered. In other words, ask yourself, “What is the largest value that divides both of these numbers?” Understanding this concept is essential for dividing and factoring polynomials.

Prime factorization can also be used to determine the greatest common factor. However, rather than multiplying all the prime factors like we did for the least common multiple, we will multiply only the prime factors that the numbers share. The resulting product is the greatest common factor.

factor trees


Review

Let’s wrap up with a couple of true or false review questions:

  1. The least common multiple of 45 and 60 is 15.
  2.  
    The answer is false. The greatest common factor of 45 and 60 is 15, but the least common multiple is 180.

  3. The least common multiple is a number greater than or equal to the numbers being considered.
  4.  
    The answer is true. The least common multiple is greater than or equal to the numbers being considered, while the greatest common factor is equal to or less than the numbers being considered.

Thanks for watching, and happy studying!

Frequently Asked Questions

Q

What are multiples and factors?

A

Factors and multiples are very closely related. Factors are the numbers we can multiply together in order to get another number. The number that is created by multiplying these factors is called a multiple. Multiples can be found by multiplying factors, or simply by skip counting. For example, the multiples of \(5\) can be found by skip counting by \(5\)s: \(5,10,15\)…

Q

What is the difference between a factor and a multiple?

A

Factors are the numbers that are multiplied in order to create a multiple. Multiples are the numbers created by multiplying two factors. Multiples are unlimited, and can be found by simply skip counting. For example, the multiples of \(30\) are \(30,60,90\)… and so on. However, the factors of \(30\) are limited. The factors of \(30\) are the numbers that multiply to \(30\), such as \(1\times30\), \(2\times15\), and \(5\times6\).

Q

What is a multiple in math?

A

A multiple can be found by skip counting, but it is also the result of multiplying a number by an integer. This integer can be positive or negative. For example, \(-6\) is a multiple of \(3\) because \(3\times-2=-6\).

Q

How do you calculate multiples?

A

Multiples can be found simply by skip counting. For example, the multiples of \(7\) can be found by skip counting by \(7\)s: \(7,14,21,28,35\)… Multiples can also be calculated by multiplying two factors. For example, \(4\) and \(5\) can be multiplied to create the product \(20\), which makes \(20\) a multiple of both \(4\) and \(5\).

Q

Why are factors and multiples important?

A

Factors and multiples are important in the field of math in many ways. For example, factors and multiples are used when looking for patterns in numbers, simplifying fractions, or when determining the greatest common factor. For example, when simplifying the fraction \(\frac{8}{10}\), we see that both \(8\) and \(10\) share a common factor of \(2\), so both numbers can be divided by \(2\) in order to simplify the fraction. \(\frac{8}{10}\) becomes \(\frac{4}{5}\).

Q

What is the definition of a factor in math?

A

Factors are the numbers we can multiply together in order to get another number. For example, the factors of \(20\) are \(1,2,4,5\) and \(20\) because these numbers can be multiplied together to get \(20\). \(1\times20=20\), \(2\times10=20\), and \(4\times5=20\). Technically, factors can be negative as well, but in general, when you are listing the factors of a number, you only need to list the positive factors.

Q

What does it mean to be a multiple?

A

In order for a number to be considered a multiple, it needs to be the result of multiplying a number and an integer. An integer is simply a whole number that is not a fraction. For example, \(36\) is a multiple of \(9\) because we can multiply \(9\) by \(4\) and the product is \(36\). It can be helpful to think about multiples as the result of skip counting. For example, \(36\) is a multiple of \(9\) because we can skip count by \(9\)s and reach \(36\): \(9,18,27,36\).

Practice Questions

Question #1:

 
What is the greatest common factor of 16 and 42? Use it to reduce the fraction \(\frac{16}{42}\).

GCF is 8, and we reduce to \(\frac{2}{5}\).

GCF is 1, and we cannot reduce any further.

GCF is 4, and we reduce to \(\frac{4}{11}\).

GCF is 2, and we reduce to \(\frac{8}{21}\).

Answer:

The correct answer is D: GCF is 2, and we reduce to \(\frac{8}{21}\).

Let’s approach this problem by listing the prime factors of both the numerator and the denominator.
\(16=2×2×2×2\)
\(42=2×3×7\)

Here we see that 2 is the only shared factor of 16 and 42 and is therefore their greatest common factor. We can then divide both numbers by 2 to reduce the fraction:
\(\frac{16\div2}{42\div2}=\frac{8}{21}\)

Question #2:

 
Find the least common multiple of 2, 6, and 8.

16

18

24

48

Answer:

The correct answer is C: 24.

For this problem, let’s list the prime factors of each number.
\(2=2\) (note that we could write \(2\times1\), but 1 is understood, or implied, and usually not necessary to write)
\(6=2\times3\)
\(8=2\times2\times2\)

Remember, when calculating the LCM of two or more numbers, we list each prime factor once that is shared by all of the numbers. Since each of our numbers has 2 as a prime factor, our LCM will also have 2 as one of its prime factors.
LCM \(=2\times\) _______

Now from the 6 we have a leftover 3, and from the 8 we have two 2’s remaining. We multiply those in to get
LCM \(=2\times3\times2\times2=24\)

Notice that even though 2, 6, and 8 are all factors of 48, the solution is not D, because 48 is not the smallest common multiple.

Question #3:

 
List the first several multiples of 3, 5, and 6 to find the least common multiple.

LCM is 15

LCM is 30

LCM is 18

LCM is 75

Answer:

The correct answer is B: LCM is 30.
The first several multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, …
The first several multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, …
The first several multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, …

As we see above, 30 is the first (least) number that 3, 5, and 6 have in common among their multiples, so the least common multiple is 30.

Question #4:

 
Courtney has 54 pieces of candy, and Trish has 36. They want to prepare goodie bags of candy for their friend Kim’s birthday party, but each bag needs to have an equal amount of candy. In order to have the most candy in each bag, with Courtney and Trish working separately, how many bags can they make, and how much candy will be in each bag?

10 bags, with 9 pieces of candy in each

9 bags, with 10 pieces of candy in each

15 bags, with 6 pieces of candy in each

5 bags, with 18 pieces of candy in each

Answer:

The correct answer is D: 5 bags, with 18 pieces of candy in each.

To begin, list the prime factors of both 54 and 36:
\(54=2\times3\times3\times3\)
\(36=2\times2\times3\times3\)

Notice that they both share a 2 and two 3’s. The product of these shared prime factors is \(2\times3\times3=18\). We now know that the GCF is 18, which means each bag will contain 18 pieces of candy. Courtney’s 54 pieces will make 3 bags, and Trish’s 36 pieces will make 2 bags. Together, they will make 5 bags with 18 pieces of candy in each.

Question #5:

 
Sara is buying fruit for an office brunch, and she needs an equal number of apples and bananas. However, the apples are sold in bags of 4 and the bananas are sold in bunches of 6. What is the least number of apples and bananas Sara can buy?

24 apples and 24 bananas

18 apples and 18 bananas

12 apples and 12 bananas

16 apples and 16 bananas

Answer:

The correct answer is C: 12 apples and 12 bananas.

With this problem, we want to know the least common multiple of 4 and 6. Using the prime factors method, we see that
\(4=2\times2\)
\(6=2\times3\)
LCM \(=2\times2\times3=12\)

Sara will buy three bags of apples and two bunches of bananas in order to have 12 of each fruit.

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by Mometrix Test Preparation | Last Updated: September 15, 2021