Unlocking the Secrets of Greatest Common Factor and Least Common Multiple with Factor Trees

Number	Smallest Prime Factor	Quotient
$80$	$2$	$\frac{8 0}{2} = 40$
$40$	$2$	$\frac{4 0}{2} = 20$
$20$	$2$	$\frac{2 0}{2} = 10$
$10$	$2$	$\frac{1 0}{2} = 5$
Prime Factorization
$80 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5$

Number

Smallest Prime Factor

Quotient

80

2

\frac{8 0}{2} = 40

40

2

\frac{4 0}{2} = 20

20

2

\frac{2 0}{2} = 10

10

2

\frac{1 0}{2} = 5

Prime Factorization

80 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5

Numbers	Multiples of Numbers	Common Multiples	Least Common Multiple
$2$ and $3$	$Multiples of 2 : Multiples of 3 : 2, 4, 6, 8, 10, 12, \dots 3, 6, 9, 12, 15, \dots$	$\underline{6}, 12, 18, 24, \dots$	$LCM (2, 3) = 6$
$8$ and $12$	$Multiples of 8 : Multiples of 12 : 8, 16, 24, 32, 40, 48, \dots 12, 24, 36, 48, \dots$	$\underline{24}, 48, 72, 96, \dots$	$LCM (8, 12) = 24$

Numbers

Multiples of Numbers

Common Multiples

Least Common Multiple

2

and

3

Multiples of 2 : Multiples of 3 : 2, 4, 6, 8, 10, 12, \dots 3, 6, 9, 12, 15, \dots

\underline{6}, 12, 18, 24, \dots

LCM (2, 3) = 6

8

and

12

Multiples of 8 : Multiples of 12 : 8, 16, 24, 32, 40, 48, \dots 12, 24, 36, 48, \dots

\underline{24}, 48, 72, 96, \dots

LCM (8, 12) = 24

Prime Factors	Multiplication	$LCM (54, 60)$
$2,$ $2,$ $3,$ $3,$ $3$ and $5$	$2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 5$	$540$

Prime Factors

Multiplication

LCM (54, 60)

2,

2,

3,

3,

3

and

5

2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 5

540

Factors	Multiplication	$LCM (10, 16)$
$2,$ $2,$ $2,$ $2,$ and $5$	$2 \cdot 2 \cdot 2 \cdot 2 \cdot 5$	$80$

Factors

Multiplication

LCM (10, 16)

2,

2,

2,

2,

and

5

2 \cdot 2 \cdot 2 \cdot 2 \cdot 5

80

A citywide high school soccer club has $100$ juniors, $88$ sophomores, and $76$ seniors. The head coach wants to divide the students into groups of the same size. Each team must have the same numbers of juniors, sophomores, and seniors. What is the greatest possible number of groups the coach can make?

Davontay, Vincenzo, and Tadeo saw each other at the cinema today. If Davontay goes to the cinema every $6^{th}$ day, Vincenzo every $1 0^{th}$ day, and Tadeo every $5^{th}$ day, how many times must Tadeo go before the three friends will meet at the cinema again?

Number	Factors
$100$	$1,$ $2,$ $4,$ $5,$ $10,$ $20,$ $25,$ $50,$ $100$
$88$	$1,$ $2,$ $4,$ $8,$ $11,$ $22,$ $44,$ $88$
$76$	$1,$ $2,$ $4,$ $19,$ $38,$ $76$

Number

Factors

100

1,

2,

4,

5,

10,

20,

25,

50,

100

88

1,

2,

4,

8,

11,

22,

44,

88

76

1,

2,

4,

19,

38,

76

Number	Multiples
$6$	$6,$ $12,$ $18,$ $24,$ $30,$ $36,$ $42,$ $\dots$
$10$	$10,$ $20,$ $30,$ $40,$ $50,$ $60,$ $70,$ $\dots$
$5$	$5,$ $10,$ $15,$ $20,$ $25,$ $30,$ $35,$ $\dots$

Number

Multiples

6

6,

12,

18,

24,

30,

36,

42,

\dots

10

10,

20,

30,

40,

50,

60,

70,

\dots

5

5,

10,

15,

20,

25,

30,

35,

\dots

Select the prime factorization of each given number.

278

We are asked to find the prime factorization of 204. We will use a factor tree to find this prime factorization. Let's recall the process for creating a factor tree.

We now know how to create a factor tree! Let's create a factor tree for 204.

We found the prime factorization of 204. Prime Factorization of204 204=2*2*3*17

We will follow a similar process to find the prime factorization of 278.

The only prime factors in the prime factorization of 278 are 2 and 139. We can now write the prime factorization corresponding to 278. Prime Factorization of278 278=2*139

Find the number that corresponds to the given prime factorization.

2 \cdot 2 \cdot 3 \cdot 5 \cdot 7

2 \cdot 2 \cdot 5 \cdot 13

3 \cdot 67

Let's begin by looking at the given prime factorization. 2*2*3*5*7 We want to determine the number represented by this prime factorization. We can use the Associative Property of Multiplication to calculate partial products. Then we will repeat this process until we get the final product.

The product of the given prime factorization is 420, which means that the prime factorization represents 420.

We will follow a similar process to calculate the number that corresponds to this prime factorization.

We can now say that this is the prime factorization of 260. 2*2*5*13 = 260

Finally, we only have a pair of prime factors. Let's calculate their product! 3*67= 201 The third prime factors represent the prime factorization of 201.

Dominika's favorite number is

9 .

Her best friend gives her a list with some pairs of numbers. He challenges Dominika to find which of the pairs in the list have Dominika's favorite number as their greatest common factor. Help Dominika choose the correct pair of numbers.

We want to know which of the given pairs has as the greatest common factor 9. Let's begin by finding the greatest common factor of each pair. Recall the process for finding the greatest common factor of two numbers.

Let's find the greatest common factor of each pair of numbers. This will help us determine the pair that has Dominika's favorite number as its greatest common factor.

74 and 49

Let's create factor trees for 74 and 49 to find their prime factorizations.

Now let's write the prime factorizations together to identify their common prime factors. 74=& 2*37 49=& 7*7 Each number has only two prime factors and there are no common factors between them. This means that the only common factor between these numbers is 1. In other words, 1 is the greatest common factor of 74 and 49. GCF( 74, 49)= 1

45 and 63

Now let's find the greatest common factor of 45 and 63.

Let's find the common prime factors between these prime factorizations. 45=3*3*5 63=3*3*7 We have that 3* 3 are common factors between the prime factorizations. The product of these is the GCF( 45, 63). GCF( 45, 63)= 3* 3 ⇒ GCF( 45, 63)= 9

92 and 76

We can use a similar process to find the prime factorizations of 92 and 76. Consider the prime factorizations of these numbers. 92=2*2*23 76=2*2*19 The product of the common factors 2* 2, and therefore the greatest common factor of 92 and 76, is 4. GCF( 92, 76)= 4

32 and 46

Finally, let's look at the prime factorizations of 32 and 46. 32=& 2*2*2*2*2 46=& 2*23 The only common prime factor between 32 and 46 is 2 — that is, the GCF( 32, 46)= 2. GCF( 32, 46) = 2

Conclusion

We found the greatest common factor for each pair of numbers in Dominika's list. Let's summarize our results in a table!

Numbers	Prime Factorization	GCF	Is 9 the GCF?
74 and 49	74=2*37	1	No
74 and 49	49=7*7	1	No
45 and 63	45=335	9	Yes
45 and 63	63=337	9	Yes
92 and 76	92=2223	4	No
92 and 76	76=2219	4	No
32 and 46	32=22222	2	No
32 and 46	46=2*23	2	No

We can see from this table that 45 and 63 are the only pair of numbers whose GCF is 9. Great! Dominika has successfully completed her friend's challenge.

Look at the following questions.

A . B . C . D . What is the highest common divisor of 24 and 36 ? What is the greatest common factor of 24 and 36 ? What is the product of the common factors in the prime factorization of 24 and 36 ? What is the greatest common prime factor of 24 and 36 ?

Which question does not belong with the other three?

We will go through each of the questions and find their answers. This will allow us to determine which question does not belong with the other three.

Question A

Let's look at the question given in option A.

What is the highest common divisor of 24 and 36?

This question asks for the highest common divisor of 24 and 36. The word highest means the greatest. Recall that a factor of a number divides that number evenly, without any remainder. This means that this question is asking for the greatest common factor of these two numbers. Let's list the factors of this pair of numbers.

	Factors
24	1, 2, 3, 4, 6, 8, 12, 24
36	1, 2, 3, 4, 6, 9, 12, 18, 36

These numbers has six common factors. The highest of these factors is 12.

Question B

Now, consider the question of option B.

What is the greatest common factor of 24 and 36?

This question also asks for the greatest common factor of 24 and 36. This means that the answer to this question is the same as the previous question. GCF( 24, 36)= 12

Question C

Next, let's analyze the question in option C.

What is the product of the common factors in the prime factorization of 24 and 36?

This time, we will calculate the product of the common factors in the prime factorization of 24 and 36. Let's first look at the prime factorizations of each. & 24= 2* 2*2* 3 & 36= 2* 2* 3*3 The common factors shared by the prime factorizations are 2, 2, and 3. Let's calculate their product. 2* 2* 3= 12 We got the same result as we did for the previous questions. This is because one way to determine the greatest common factor of two numbers is to multiply the common factors in their prime factorizations!

Question D

Finally, let's analyze the last question given in option D.

What is the greatest common prime factor of 24 and 36?

We are asked this time for the greatest prime common factor of 24 and 36. Let's consider the prime factorizations of the numbers again. & 24=2*2*2* 3 & 36=2*2* 3*3 The greatest common prime factor is 3. The answer for this question is 3.

Conclusion

We analyzed the four given questions. Let's summarize our results in a table to determine which question does not belong.

	Question	Answer
A	What is the highest common divisor of 24 and 36?	12
B	What is the greatest common factor of 24 and 36?	12
C	What is the product of the common prime factors of 24 and 36?	12
D	What is the greatest common prime factor of 24 and 36?	3

We can now conclude that the question that does not belong with the other three is option D.

Ramsha used number lines to describe the multiples of some pair of numbers. However, she is having difficulty understanding these figures.

Number lines representing different multiples

Help Ramsha determine which of the models represents a least common multiple that is different from the others.

Let's analyze the given models to identify the multiples they describe. We will then have a better idea of the least common multiple each represents, which will help us to determine which model is different from the other three.

Option A

Consider the model A.

Each path in this model describes the multiples of a certain number. Notice that the red path goes from 9 to 18, then to 27, and so on, until it reaches 36. This means we can say that the red path describes the multiples of 9. Multiples of9: 9, 18, 27,36 Let's consider the blue path. We can see that it increases by 12 for every jump. This means that this path describes multiples of 12, starting with 12 and going until it reaches 36. Multiples of12: 12, 24, 36 We found the multiple that each path describes. We can see from the list of multiples created that the least common multiple of 9 and 12 is 36. This corresponds with the point where both paths touch the number line together.

Option B

Now let's look at model B.

In this model, the red path increases by 18 each jump. This means that it describes the multiples of 18. The blue path increases by 12 on each step, so it describes the multiples of 12. Let's look at the meeting point of this model.

The meeting point of the paths is also 36. This means that the least common multiple described by this model is also 36.

Option C

Next, consider model C.

The red path in this model describes the multiples of 4 because it increases by 4 on every step. The blue path describes the multiples of 12 because it jumps by 12 every step. The first point where the lines meet, which represents the least common multiple of these numbers, is 12.

Option D

Finally, let's look at model D.

The red path describes the multiples of 4, while the blue path describes the multiples of 9. The LCM( 4,9) is shown to be 36.

Conclusion

We analyzed all the given models. Let's make a table summarizing the results!

Model	Multiples of	Least Common Multiple
A	9 and 12	36
B	18 and 12	36
C	4 and 12	12
D	4 and 9	36

We can see that models A, B, and C describe pairs of numbers whose least common multiple is 36. Conversely, model C describes a pair of numbers whose least common multiple is 12. This means that model C is the one that does not belong.

Calculate the greatest common factor for each pair of numbers.

51, 48

74, 80

One method to find the greatest common factor of two numbers is to list the factors of each number. The next step is to identify the common factors in this list. The greatest of these common factors represents the greatest common factor of the numbers. Let's create a list of all the factors of 51 and 48. Factors of51:& 1, 3, 17, 51 [0.5em] Factors of48:& 1, 2, 3, 4, 6, 8, 12, 16, & 24, 48 We can see that the common factors between these numbers are 1 and 3. The greater of these common factors is 3, so the greatest common factor of 51 and 48 is 3.

We will follow a similar process to find the greatest common factor of 74 and 80. We will start by creating a list of the factors of these numbers. Factors of74:& 1, 2, 37, 74 [0.5em] Factors of80:& 1, 2, 4, 5, 8, 10, 16, 20, & 40, 80 The common factors of these numbers are 1 and 2. Since 2 is greater than 1, this means that the GCF( 74, 80)= 2.

Find the least common multiple of the given pair of numbers.

22, 17

15, 10

We want to calculate the least common multiple (LCM) of 22 and 17. Let's begin by creating a factor tree for both numbers.

We have now the prime factorization of this pair of numbers. Let's write them together to see the different prime factors. & 22=2*11 & 17=17 The prime factorizations have three different factors that all appear only once. This means that we must multiply these three factors to get the LCM( 22, 17). Let's do it! LCM( 22, 17)=2*11*177 ⇕ LCM( 22, 17)=374

We want to find the least common multiple of 15 and 10. This time, we will create a list containing the first few multiples of each number. Then we will identify the common multiples between the lists. The least of these will be the least common multiple of 15 and 10. Let's do it! &Multiples of15: 15, 30, 45, 60, 75, 90,... &Multiples of10: 10, 20, 30, 40, 50, 60,... We found that the first two common multiples of 15 and 10 are 30 and 60. The smaller of these — and therefore the least common multiple — is 30. LCM( 15, 10)= 30

	16 Theory slides
	16 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Greatest Common Factor and Least Common Multiple

Catch-Up and Review

Hint

Solution

Hint

Solution

Prime Factorization of 35

Prime Factorization of 84

Identify Common Prime Factors

Multiply

Hint

Solution

Prime Factorization of 10 and 16

Math Prime Factors Vertically

Bring Down the Primes in Each Column and Multiply

Finding the Numbers to Fill in the Orange and Pink Cells

Hint

Solution

Checking Our Answer

Hint

Solution

74 and 49

45 and 63

92 and 76

32 and 46

Conclusion

Question A

Question B

Question C

Question D

Conclusion

Option A

Option B

Option C

Option D

Conclusion

Greatest Common Factor and Least Common Multiple

Recommended exercises

Prime Factorization of $35$

Prime Factorization of $84$

Prime Factorization of $10$ and $16$