Factorization and Greatest Common Divisor (GCD

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© FEB2014 by Daniel’sMathTutorials
Introductionto
Factorization
The Greatest Common Divisor (GCD), also known as the Highest Common Factor (HCF),
is the biggest number that can divide two or more numbers.
In this tutorial, we will consider the concept of factors, or divisors. We will then discuss
prime numbers, and how to express a number that is not a prime number as a product of
prime numbers.
Finally, we will consider the concept of common factors, and most importantly, how to
find the HCF of two or more numbers.
However, in order to gain mastery of the methods presented here, practice questions are
provided at the end of every section.
Contents
1. Factors 1
2. Factorization 3
3. Prime Numbers 7
4. Prime Factors 10
5. Prime Factorization 13
6. Common Factors 18
7. Highest Common Factors 19
Answers to Exercise Questions 22
1. Factors
Division of one whole number by another is a very familiar arithmetic operation. When
dividing one whole number by another whole number, another whole number could be
the result.
For instance, 5 divides 40 eight times exactly. (Or, 40 = 5 × 8)
Also, 4 divides 40 ten times exactly. (Or, 40 = 4 × 10)

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Since 5 and 4 divide 40 exactly, without leaving remainders, both of them are called
Factors, or Divisors, of 40.
However, 6 is not a factor of 40 because
40 ÷ 6 = 6 times and 4 remainders
For the same reason, 11 is also not a factor of 40, because
40 ÷ 11 = 3 times and 7 remainders
Definition
Let x stand for any positive whole number
A Factor of x divides x without leaving a remainder.
Note
1 can divide every other number without leaving remainders. So 1
is a factor of every number.
Note
Since every number can divide itself exactly, every number is a
factor of itself.
The examples below show how to determine if a number is a factor of another
Example
Say whether 3, 4, and 5 are factors of 24. Give reasons.
3 is a factor of 24
Reason: 24 contains 3 eight times and 0 remainder
OR, 24 = 3 × 8
4 is a factor of 24
Reason: 24 contains 4 six times and 0 remainder
OR, 24 = 4 × 6
5 is not a factor of 24
Reason: 24 contains 5 four times and 4 remainders
OR, 24 = (5 × 4) + 4

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Example
Say whether 10, 14, and 16 are factors of 60. Give reasons.
10 is a factor of 60
OR, 60 = 10 × 6
Reason: 60 contains 14 four times and 4 remainders
OR, 60 = (14 × 4) + 4
Reason: 60 contains 16 three times and 12 remainders
OR, 60 = (16 × 3) + 12
Exercise 1
1. Which of these are factors of 75?
(i) 2 (ii) 3 (iii) 5 (iv) 15
2. Which of these are NOT factors of 64?
(i) 12 (ii) 8 (iii) 18 (iv) 16
3. Say whether 7, 9 and 15 are factors of 45. Give reasons.
4. Say whether 10, 12 and 14 are factors of 70. Give reasons.
2. Factorization
It is possible to write a positive whole number, say 40, as a product of its factors.
For instance, 40 = 8 × 5 or 5 × 8
Notice that 5 divides 40 eight times, and 8 divides 40 five times.
Writing 40 as 5 × 8 is called Factoring, or Factorizing, 40 because we have now written
40 as a multiplication of two of its factors.
40 can also be factored, or factorized, in other ways, like
40 = 1 × 40 or 40 × 1
40 = 4 × 10 or 10 × 4
e.t.c.

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Definition
To Factorize a whole number means to write that number as a
product, or a multiplication, of its factors.
Note
Saying “Factorize a number” and “Factor a number” both mean
the same thing.
The examples below show how to determine all the factors of a given number.
Example
Find all the factors of 16.
First find all the possible ways you can express 16 as a product of a pair of its
factors
1 is a factor of every number, so
16 = 1 × 16
Next try 2: 2 is also a factor of 16, 16 contains 2 eight times
16 = 1 × 16
2 × 8
Next try 3: 3 is not a factor of 16
Next try 4: 4 is a factor of 16, 16 contains 4 four times
16 = 1 × 16
2 × 8
4 × 4
Next try 5, then 6, then 7, until you get to 16
16 = 1 × 16
2 × 8
4 × 4
8 × 2 (Same as 2 × 8)
16 × 1 (Same as 1 × 16)

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When we remove the repetitions
2 × 8 and 8 × 2,
1 × 16 and 16 × 1,
We get all the possible ways to factorize 16 as a pair of two factors
16 = 1 × 16
2 × 8
4 × 4
Hence, all the factors of 16 are: 1, 2, 4, 8 and 16
Example
Find all the factors of 60.
Since 1 can divide 60 1 is a factor of 60
60 = 1 × 60
Next try 2: 2 is also a factor of 60, 60 contains 2 thirty times
60 = 1 × 60
2 × 30
Next try 3: 3 is also a factor of 60, 60 contains 3 twenty times
60 = 1 × 60
2 × 30
3 × 20
Next try 4 4 is a factor of 60, 60 contains 4 fifteen times
60 = 1 × 60
2 × 30
3 × 20
4 × 15
follow the arrow

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Next try 5 5 is a factor 60 contains 5 twelve times
60 = 1 × 60
2 × 30
3 × 20
4 × 15
5 × 12
Next try 6 6 is also a factor 60 contains 6 ten times
60 = 1 × 60
2 × 30
3 × 20
4 × 15
5 × 12
6 × 10
Between 6 and 10, there is no factor of 60.
The next factor of 60 is 10, which is already there.
After 10, the next factor is 12. You can see that 12 is already there.
So we finished all the possibilities.
60 = 1 × 60
2 × 30
3 × 20
4 × 15
5 × 12
6 × 10
The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
Exercise 2
Find all the factors of
(a) 24 (b) 36 (c) 54 (d) 100
follow the arrow

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3. Prime Numbers
Let us list the numbers from 1 to 15, along with their factors
Number Factors Number Factors
1 1 9 1, 3, 9
2 1, 2 10 1, 2, 5, 10
3 1, 3 11 1, 11
4 1, 2, 4 12 1, 2, 3, 4, 6, 12
5 1, 5 13 1, 13
6 1, 2, 3, 6 14 1, 2, 7, 14
7 1, 7 15 1, 3, 5, 15
8 1, 2, 4, 8
From the above table, we can see that some numbers have only two factors: themselves
and 1. This simply means that the only numbers that can divide them are themselves and
1.
The first two of such numbers are 2 and 3. Can you locate the others?
Right! The rest are 5, 7, 11 and 13.
Such numbers are called Prime Numbers
Definition
A Prime Number is a number that has only two factors, itself
and 1.
Note
1 is not a prime number. This is because it has only one factor;
itself.
From the table, we can see that we have 6 prime numbers between 1 and 15: 2, 3, 5, 7, 11
and 13.
If we wanted to get the prime numbers, say up to 100, it would be tedious to get all
factors for each number, check if they are only two factors for the number, and conclude
that the number is prime number.

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However, there is another more convenient method, called the Sieve of Eratosthenes,
which is described below:
 List the whole numbers from 1 to the number you want to stop at, say 70, or maybe
100
 Strike out 1, since it has only one factor: itself. So it is not prime
 2 is a prime. Leave it. Strike out every multiple of 2 beginning from 4. (4 = 2 ×
2)
 The first number not struck out by 2 is 3. So 3 is a prime. Leave it. Strike out every
multiple of 3 beginning from 9. (9 = 3 × 3)
 The first number not struck out by 3 is 5. So 5 is also a prime. Leave it. Strike out
every multiple of 5 beginning from 25. (25 = 5 × 5)
 The first number not struck out by 5 is 7. So 7 is a prime. Leave it. Strike out every
multiple of 7 beginning from 49. (49 = 7 × 7)
And so on until there are no more numbers to strike out. The remaining numbers are the
prime numbers in the range of numbers you chose.
The example below demonstrates how to use this method to find all the prime numbers
between 1 and 50.
Example
Use the Sieve of Eratosthenes method of finding prime numbers to find all prime
numbers between 1 and 50.
List the numbers from 1 to 50
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50

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Strike out 1; 1 is not a prime number. (Each number I strike out will be shaded)
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
The first prime number is 2. Leave it. Strike out every multiple of 2 starting with 2
× 2 = 4
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
3 is the first whole number that is not a multiple of 2. So it is prime, leave it. But
strike out every multiple of 3 starting with 3 × 3 = 9
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
5 is the first whole number that was not struck out by 3. So 5 is prime. Strike out
every multiple of 5 starting with 5 × 5 = 25
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
The next prime number is 7 (2, 3 or 5 did not succeed in striking it out)

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Strike out all multiples of 7 starting from 7 × 7 = 49
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
The next prime number is 11, (For the same reasons as before). 11 × 11 = 121 is not in
the grid. So we can stop.
The remaining numbers are the prime numbers between 1 and 50:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47.
Exercise 3
1. In the introduction, the only prime numbers we had were 2, 3, 5, 7, 11 and 13. The
example above produced prime numbers up to 47. Try to show that the extra
numbers produced are prime numbers by finding their factors.
2. Find all the prime numbers between 1 and 100.
4. Prime Factors
If we were to list all the factors of 28, we would get the following six factors:
28 = 1 × 28
2 × 14
4 × 7
That is: 1, 2, 4, 7, 14, 28
Notice that out of the six factors of 28, 2 and 7 are prime numbers. These are the prime
factors of 28.

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Also, listing the eight factors of 30, we get
30 = 1 × 30
2 × 15
3 × 10
5 × 6
That is: 1, 2, 3, 5, 6, 10, 15, 30
Here there are also prime numbers: 2, 3 and 5. These are the prime factors of 30
Definition
Let x stand for any positive whole number
A Prime Factor of x is a prime number which is also a factor of x.
Note
Not all factors of a number are prime numbers. However, if a
factor is also a prime number, it is called a prime factor.
Here are some examples
Example
Find the prime factors of 42.
First Work out the factors of 42
Working:
The factors of 42: 1, 2, 3, 6, 7, 14, 21 and 42.
Then get those factors which are prime numbers
Prime factors of 42: 2, 3, 7
42 = 1 × 42
2 × 21
3 × 14
6 × 7

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Example
Working
The factors of 24: 1, 2, 3, 4, 6, 8, 12 and 24.
Prime factors: 2 and 3
Example
Working
The factors of 75: 1, 3, 5, 15, 25, 75
Prime factors: 3 and 5
Example
29 is a prime number
So, the factors of 29 are just 1 and 29
And the prime factors of 29? Just 29
Exercise 4
Find the prime factors of
(a) 20 (b) 37 (c) 50 (d) 91 (e) 120
24 = 1 × 24
2 × 12
3 × 8
4 × 6
75 = 1 × 75
3 × 25
5 × 15

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5. Prime Factorization
Every number has one or more prime factors, as seen in the previous section. In other
words, for every whole number x, there is at least one prime number that can divide x.
What is the implication of this? For example consider 450. Take any pair of factors of
450:
450 = 45 × 10
Obviously, these are not prime numbers. Try it again for both 45 and 10
45 = 5 × 9, and 10 = 2 × 5
For both factors of 45, only 5 is a prime factor. For the pair of factors of 10, both 2 and 5
are prime factors of 10. So
450 = 45 × 10 = 5 × 9 × 2 × 5
Since 9 is not a prime number, 9 = 3 × 3
So finally, this is what we get
450 = 45 × 10 = (5 × 9) × (2 × 5)
= 5 × (3 × 3) × 2 × 5
= 5 × 3 × 3 × 2 × 5
We cannot factorize the last expression further because each of them would give
something like this
5 × 3 × 3 × 2 × 5 = (5 × 1) × (3 × 1) × (3 × 1) × (2 × 1) × (5 × 1)
The reason being that each factor in the multiplication is a prime number
Notice we now have 450 as a product of prime numbers; 2, 3 and 5.
2, 3 and 5 are prime factors of 450 (check this as an exercise)

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But a natural question might arise:
If I were to find all the factors of 450, I get more than one product
450 = 1 × 450
2 × 225
3 × 150
5 × 90
6 × 75
9 × 50
10 × 45
15 × 30
18 × 25
Is there also more than one way to write 450 as a product of its prime factors?
To find out, let us carry out a little experiment
 450 = 45 × 10 = (5 × 9) × (2 × 5) = 5 × (3 × 3) × 2 × 5 = 5 × 3 × 3 × 2 × 5
 450 = 9 × 50 = (3 × 3) × (5 × 10) = 3 × 3 × 5 × (2 × 5) = 3 × 3 × 5 × 2 × 5
 450 = 18 × 25 = (2 × 9) × (5 × 5) = 2 × (3 × 3) × 5 × 5 = 2 × 3 × 3 × 5 × 5
 450 = 3 × 150 = 3 × 10 × 15 = 3 × (2 × 5) × (3 × 5) = 3 × 2 × 5 × 3 × 5
Notice that no matter which product we started with, the final product of prime factors is
always a rearrangement of the product 2 × 3 × 3 × 5 × 5
This is actually a result in Mathematics called The Fundamental Theorem of
Arithmetic.
The Fundamental Theorem of Arithmetic
Stated informally; if a whole number x is not a prime number, we
can always write it as a product of prime numbers in only one way
(if we ignore any rearrangement of the factors). The product
would contain only the prime factors of x.
The following examples will illustrate this.

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Example
Express 180 as a product of its prime factors.
180 = 2 × 90
= 2 × 2 × 45
= 2 × 2 × 3 × 15
= 2 × 2 × 3 × 3 × 5
Example
550 = 2 × 275
= 2 × 5 × 55
= 2 × 5 × 5 × 11
Alternatively, we could use short division repeatedly to solve each of the above problems.
This method is also called Trial Division.
Trial Division requires knowledge of the prime numbers.
The next set of examples shows how to use trial division to arrive at a prime factorization
of whole numbers.
Example
Write 180 as a product of its prime factors using trial division.
Try dividing 180 by the first prime number, 2. Since 2 is a prime factor of 180, we
have
2 180
90
Try dividing 90 by 2 again
2 180
2 90
45

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2 is not a factor of 45. Then try 45 divided by the next prime number 3. Since 3 is a
prime factor of 45, we have
2 180
2 90
3 45
15
Now we have 15. If we try 3 again, 3 can still divide 15.
2 180
2 90
3 45
3 15
5
The only number that can divide 5 is 5. (Remember, 5 is a prime number) So divide
by 5 to complete the process
2 180
2 90
3 45
3 15
5 5
1
Example
For 550, the only prime numbers that will divide are 2, 5, and 11 (the prime factors
of 550)
First try 2:
2 550
275
2 × 2 × 3 × 3 × 5 = 180

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Try 5:
2 550
5 275
55
Try 5 again:
2 550
5 275
5 55
11
Finally, try 11:
2 550
5 275
5 55
11 11
1
Example
Use Trial division to find the prime factorization of 600.
600 = 2 × 2 × 2 × 3 × 5 × 5
2 600
2 300
2 150
3 75
5 25
5 5
1
2 × 5 × 5 × 11 = 550

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Example
Use Trial division to find the prime factorization of 588.
588 = 2 × 2 × 3 × 7 × 7
Exercise 5
Write the following numbers as a product of their respective prime factors
(b)20 (b) 30 (c) 12 (d) 66 (e) 78
6. Common Factors
In the previous sections, we discussed how to find the factors of a single whole number.
So, if a number divides x without leaving a remainder, that number is a factor of x. For
example, 1, 2, 3, 4, 6 and 12 are all factors of 12 because they divide 12 without leaving a
remainder.
Now to something slightly different: Factors common to two or more numbers. Say, you
are given two numbers 12 and 18. Of course 1 can divide both 12 and 18. So 1 is a common
factor of 12 and 18. We also have 2 as a common factor of 12 and 18. Actually, all the
common factors of 12 and 18 are 1, 2, 3, and 6.
Definition
A Common Factor of two or more numbers divides the two
numbers without leaving any remainders.
The following examples show how to find all the Common factors of two or more
numbers.
2 588
2 294
3 147
7 49
7 7
1

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Example
Get all the common factors of 18 and 24.
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Common factors of 16 and 18: 1, 2, 3, 6
Example
Get all the common factors of 16, 24, and 40.
Factors of 16: 1, 2, 4, 8, 16
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
Common factors of 16, 24, 40: 1, 2, 4, 8
Exercise 6
List all the common factors of the following
(a) 18 and 27 (b) 12 and 30 (c) 15, 24, and 33
7. Highest Common Factors
Consider the following example
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 45: 1, 3, 5, 9, 15, 45
Common factors of 30 and 45: 1, 3, 5, 15
Four factors are common to 30 and 45. But 15 is the largest number that can divide both
30 and 45. So 15 is said to be the highest common factor of 30 and 45.
Definition
The Highest Common Factor of two numbers is the largest
whole number that is a common factor of the two numbers.
But the above method for finding the HCF of 30 and 45 is a little bit tedious. It is possible
to use the prime factorization method to find the HCF of 30 and 45.
The following examples will illustrate this.

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Example
Use the prime factorization method to find the HCF of 30 and 45
In this method, first express 30 and 45 each as products of their prime factors. Next,
carefully observe which “smaller product” in 30 is the same with a “smaller product” in
45. The smaller product will be equal to the HCF of 30 and 45.
2 30
3 15
5 5
1
3 45
3 15
5 5
1
Example
Find the HCF of 16 and 28.
16 = 2 × 2 × 2 × 2
28 = 2 × 2 × 7
HCF = 2 × 2 = 4
Example
Find the HCF of 12, 18, and 30.
12 = 2 × 2 × 3
18 = 2 × 3 × 3
30 = 2 × 3 × 5
HCF = 2 × 3 = 6
HCF = 3 × 5 = 15
30 = 2 × 3 × 5
45 = 3 × 3 × 5

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Example
Find the HCF of 20 and 21.
20 = 2 × 2 × 5
21 = 3 × 7
HCF = (Hmnh … nothing) just say 1
Why?
Let’s verify using the “long” method
Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 21: 1, 3, 7, 21
Common factors of 20 and 21: 1
The HCF: 1
Exercise 7
Find the HCF of the following
(a) 28 and 70 (b) 24 and 36 (c) 27 and 90 (d) 156, 117 and 195

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Answers to Exercise Questions
Exercise 1
(a) 3, 5 and 15 are factors of 75 because
75 = 3 × 25
75 = 5 × 15
75 = 15 × 5
(b)12 is NOT a factor of 64 because it leaves 4 remainders when dividing 64.
18 is also NOT a factor of 64 because it leaves 10 remainders after
dividing 64.
(c) 7 is not a factor of 45
OR, 45 = (7 × 6) + 3
9 is a factor of 45
Reason: 45 contains 9 five times and 0 remainders
OR, 45 = 9 × 5
Reason: 45 contains 15 three times and 0 remainders
OR, 45 = 15 × 3
(d)10 is a factor of 70
Reason: 70 contains 10 seven times and 0 remainder
OR, 70 = 10 × 7
OR, 70 = (12 × 5) + 10
OR, 70 = 14 × 5

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Exercise 2
(a) Factors of 24 = 1, 2, 3, 4, 6, 8, 12 and 24
(b)Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18 and 36
(c) Factors of 54 = 1, 2, 3, 6, 9, 18, 27 and 54
(d) Factors of 100 = 1, 2, 4, 5, 10, 20, 25, 50 and 100
Exercise 3
(2) 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.
Exercise 4
(a) Prime Factors of 20 = 2 and 5.
(b)Prime Factors of 37 = 37 only.
(c) Prime Factors of 50 = 2 and 5.
(d) Prime Factors of 91 = 7 and 13.
(e) Prime Factors of 120 = 2, 3 and 5.
Exercise 5
(a) 20 = 2 × 2 × 5
(b)30 = 2 × 3 × 5
(c) 12 = 2 × 2 × 3
(d) 66 = 2 × 3 × 11
(e) 78 = 2 × 3 × 13
Exercise 6
(a) Common Factors of 18 and 27 = 1, 3 and 9
(b)Common Factors of 12 and 30 = 1, 2, 3 and 6
(c) Common Factors of 15, 24 and 33 = 1 and 3
Exercise 7
(a) HCF of 28 and 70 = 2 × 7 = 14
(b)HCF of 24 and 36 = 2 × 2 × 3 = 12
(c) HCF of 27 and 90 = 3 × 3 = 9
(d)HCF of 156, 117 and 195 = 3 × 13 = 39

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