Factors and Multiples

Question 1

the numbers we multiply together to obtain another number (or the product). For example, when 9 and 5 are multiplied together, the result is 45.

Answer 1

factors

Question 2

Example 1: What are the factors of 21?

Answer 2

Solution: Note that:

1 x 21 = 21
3 x 7 = 21
Thus, 1, 3, 7, and 21 are factors of 21.

Question 3

Example 2: What are the factors of 150?

Answer 3

Solution: Note that:

1 x 150 = 150
2 x 75 = 150
3 x 50 = 150
5 x 30 = 150
6 x 25 = 150
And so on…

Therefore, 1, 2, 3, 4, 5, 6, 25, 30, 50, 75, and 150 are some of the factors of 150.

Question 4

Are negative numbers a factor of a number?

Answer 4

Yes, they are! For example, – 3 × – 2 = 6. Since when you multiply – 3 by -2 the result is 6, then – 3 and – 2 are factors of 6. Hence, a negative number can also be a factor.

Question 5

What is a prime number?

Answer 5

A prime number is a whole number greater than 1 that has two factors only: 1 and itself. For example, 3 is a prime number since it has only two factors which are 1 and itself (i.e., 3). You cannot think of other factors of 3 aside from 1 and 3.

Question 6

What is a composite number?

Answer 6

A composite number is a whole number that has more than two factors. For example, 6 is a composite number since it has more than two factors which are 1, 2, 3, and 6.

Remember that 1 is neither a prime number nor a composite number.

Question 7

Example: Which of the following numbers are prime?

2, 10, 19, 145

Answer 7

Solution:

Let us list all factors of each given number and determine whether they are prime or not:

For 2:

2 x 1 = 2
Factors of 2: 1, 2

Since 2 has only two factors which are 1 and itself, then 2 is a prime number.

For 10:

1 x 10 = 10
2 x 5 = 10
Factors of 10: 1, 2, 5, 10

Since 10 has more than two factors, then 10 is not a prime number.

For 19:

1 x 19 = 19
Factors of 19: 1, 19

Since 19 has only two factors which are 1 and itself, then 19 is a prime number.

For 145:

1 x 145 = 145
5 x 29 = 145
Factors: 1, 5, 29, 145

Since 145 has more than two factors, then 145 is not a prime number.

Thus, out of the given numbers, only 2 and 19 are the prime numbers.

Question 8

the process of expressing a composite number as a product of its prime factors.

Answer 8

Prime factorization

Question 9

a diagram that can be used to find the factors of any number, then the factors of those numbers, and so on until we can’t factor anymore. The ends of the factor tree are all of the prime factors of the original number.

Answer 9

factor tree

Question 10

Let’s try to determine the prime factorization of 24 using the factor tree.

Answer 10

12 and 2 are factors of 24. Notice that 2 has no factors anymore aside from 1 and itself so we stop there. On the other hand, we can still look for the factors of 12.

6 and 2 are factors of 12. We can still look for factors of 6 since it is a composite number.

3 and 2 are factors of 6. Note that both 3 and 2 do not have factors aside from 1 and themselves (they are prime numbers) so we can already stop with these numbers.

We collect all the ends of the tree (numbers that are colored red) as they are the prime factors of 24. Finally, we express 24 as a product of these prime factors.

Question 11

Example 1: Determine the prime factorization of 36 using a factor tree.

Answer 11

Start by thinking of any factors of 36. In this case, let us use 18 and 2 since 18 x 2 = 36.

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2 is a prime number so we stop on that part. However, 18 is a composite number. This means that we can still factor it out. We think again of any factors of 18. This time let us use 9 and 2 since 9 x 2 = 18.

factors and multiples 4
9 is a composite number so we can still factor it out. We think again of any factors of 9. This time, let us use 3 and 3 since 3 x 3 = 9.

factors and multiples 5
We collect all the ends of our factor tree (red-colored numbers). These are the prime factors of 36. We express 36 as a product of these prime factors.

factors and multiples 6
Therefore, the prime factorization of 36 is 3 x 3 x 2 x 2 = 36, or if written with exponents, 32 x 22 = 36

By now, you have already learned what factors, prime numbers, composite numbers, and prime factorization are. In our next section, we will apply these concepts to determine the Greatest Common Factor of two numbers.

Question 12

the largest common factor of given numbers.

Answer 12

GCF

Question 13

Example 1: Determine the GCF of 8 and 12.

Answer 13

Solution: Let us list all of the factors of 8 and 12:

Factors of 8: 1, 2, 4, 8
Factors of 12: 1, 2, 3, 4, 6, 12
By looking at our list above, the common factors are 1, 2, and 4.

However, GCF must be the largest common factor. We already know that the common factors are 1, 2, and 4. Which of these common factors is the largest? Obviously, it’s 4.

Thus, the GCF of 8 and 12 is 4.

Question 14

Example 2: What is the GCF of 15 and 25?

Answer 14

Solution: Let us list all of the factors of 15 and 25:

Factors of 15: 1, 3, 5, 15
Factors of 25: 1, 5, 25
From the list above, it is clearly seen that the only common factor is 5. Therefore, it is also the largest common factor or GCF of 15 and 25.

Thus, the GCF of 15 and 25 is 5.

Question 15

To find the GCF of given numbers using the prime factorization method, you can follow these steps:

Answer 15

Determine the prime factorization of the given numbers (i.e., use factor trees).
Express the numbers as product of their prime factors.
Match the prime factors vertically.
Bring down the common prime factors for each column. Do not bring down those in the columns where the factors are not the same.
Multiply the numbers you brought down. The result is the GCF.

Question 16

Example: Use the prime factorization method to find the GCF of 54 and 108

Answer 16

Solution:

Determine the prime factorization of the given numbers (i.e., use factor trees).
We apply the technique we discussed in the previous section to perform the prime factorization of 56 and 108.

factors and multiples 7
Express the numbers as a product of their prime factors.
From the factor trees we created above, we can express 56 and 108 as products of their prime factors. We rearrange the factors so that the same factors are vertically aligned.

factors and multiples 8
Match the prime factors vertically.
We match the common prime factors vertically using blue rectangles. We will not put a blue rectangle on a column where the prime factors are not the same.

factors and multiples 9
Bring down the common prime factors for each column. Do not bring down those in the columns where the factors are not the same.
factors and multiples 10
Multiply the numbers you brought down. The result is the GCF.
factors and multiples 11
Therefore, using the prime factorization method, the GCF of 54 and 108 is 4.

Question 17

a result when we multiply a number by an integer.

Answer 17

MULTIPLE

Question 18

Example 1: Provide five multiples of 10.

Answer 18

Solution: To find five multiples of 10, we just need to multiply 10 by five different integers. For example:

10 × 1 = 10

10 × 3 = 30

10 × – 5 = – 50

10 × – 8 = – 80

10 × 140 = 1400

Hence, 10, 30, – 50, – 80, and 1400 are some of the multiples of 10.

Question 19

Example 2: What are the first three positive multiples of 4?

Answer 19

Solution: Since we are now required to find the first three positive multiples of 4, we need to multiply 4 by the first three positive integers:

4 × 1 = 4

4 × 2 = 8

4 × 3 = 12

Thus, the first three positive multiples of 4 are 4, 8, and 12

Question 20

Example 3: How many multiples of 5 are there between 14 and 21?

Answer 20

Solution: Note that the numbers 15 and 20 which are between 14 and 21 are multiples of 5 since

5 × 3 = 15 and 5 × 4 = 20. That is, 15 and 20 are the results when 5 is multiplied by integers 3 and 4, respectively.

Thus, between 14 and 21, there are two multiples of 5.

Question 21

the smallest whole number that is a multiple of two or more numbers.

Answer 21

LCM

Question 22

Can you tell from the list which are the common multiples of 2 and 3?

Answer 22

Solution: Clearly, the common multiples of 2 and 3 are 6 and 12.

Now, you might notice that 6 is the smallest common multiple between 2 and 3. Therefore, 6 is the Least Common Multiple of 2 and 3.

Based on our example above, the Least Common Multiple is the smallest common multiple between given numbers.

Question 23

How to find the LCM using the listing method

Answer 23

One of the common ways to find the LCM of two numbers is by listing the factors of the given numbers. We already used this method for our example above through which we were able to find the LCM of 2 and 3. Now, let’s try to use this method again to find the LCM of 5, 10, and 25.

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Determining the LCM using the listing method seems to be a cakewalk. But, what if I ask you to find the LCM of 130 and 300? The listing method seems to be inconvenient in this case.

When the listing method becomes a tedious way of finding the LCM, use the prime factorization method instead.

Question 24

How to find the LCM using prime factorization

Answer 24

To find the LCM of given numbers using prime factorization, follow these steps:

Determine the prime factors of the given numbers (using the factor tree).
Express the given numbers as the product of their prime factors.
Match the prime factors vertically.
Bring down the prime factors in each column.
Multiply the factors to obtain the LCM

Question 25

Example: Determine the LCM of 130 and 300

Answer 25

1. Determine the prime factors of the given numbers (using the factor tree).

factors and multiples 14
The ends (red-colored numbers) of the respective factor trees of 130 and 300 are their prime factors. We will use these prime factors for our next step.

2. Express the given numbers as the product of their prime factors.

factors and multiples 15
3. Match the prime factors vertically.

factors and multiples 16
We match common prime factors vertically using the red rectangles. If a prime factor has no “partner” to the other number, we just leave the space blank.

4. Bring down the prime factors in each column.

factors and multiples 17
Bring down the common prime factors in each column. In our illustration above, we bring down every common prime factor on each red rectangle.

5. Multiply the factors to obtain the LCM.

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The last step is to multiply the common prime factors we have brought down from step 4. The resulting number is the Least Common Multiple (LCM).

Therefore, the LCM of 130 and 300 is 3900.

Question 26

1) Which of the following is NOT a factor of 81?
a) 3
b) 27
c) 9
d) None of the above

Answer 26

D

Question 27

2) What is the Greatest Common Factor of 35 and 70?
a) 35
b) 70
c) 15
d) 7

Answer 27

A

Question 28

3) Suppose that m and n are positive whole numbers such that m x n = 450. Which of the
following are the possible values of m?
a) 90
b) 45
c) 15
d) All of the above

Answer 28

D

Question 29

4) What is the Least Common Multiple of 18 and 54?
a) 36
b) 18
c) 54
d) 108

Answer 29

C

Question 30

5) What is the smallest even number that is also a prime number?
a) 0
b) 1
c) 2
d) 4

Answer 30

C