Composite numbers
and
Prime numbers
are two types of numbers that differ based on their factors. Factors are the numbers that divide a given number evenly without leaving a remainder.
For example, the factors of 6 are 1, 2, 3, and 6, because all these numbers divide 6 without leaving any remainder.
Prime numbers have exactly two factors: 1 and the number itself. For example, 5 is a prime number because its only factors are 1 and 5.
In this blog post, we will discuss composite numbers in detail.
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What are Composite Numbers?
Composite numbers are positive integers that can be formed by multiplying two smaller positive integers. These numbers have more than two factors, meaning they can be divided evenly by at least three different numbers: 1, the number itself, and at least one other positive integer.
For example, 4 is a composite number because it has three factors: 1, 2, and 4. Similarly, 6 is composite because its factors are 1, 2, 3, and 6. Numbers like 8, 9, and 10 are also composite, as they each have multiple divisors.
Perimeter of Rectangle
It’s important to note that 1 is not a composite number because it has only one factor: itself. Composite numbers can also include highly composite numbers, which have a particularly large number of factors.
For instance, 12 is a highly composite number because its factors are 1, 2, 3, 4, 6, and 12. Understanding composite numbers involves recognizing that they are built from smaller numbers and have multiple divisors.
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Steps to Check if a Number is Composite
To determine if a number is composite, follow the steps discussed below:
Step 1
: Find all the factors of the given number:
Step 2:
Count the total number of factors:
If the number has more than two factors, it is composite.
If it has only two factors (1 and the number itself), it is prime.
If the number has only one factor (itself), it is neither prime nor composite (e.g., 1).
For Example:
Check if 9 is composite:
Step 1
: Find factors of 9:
The factors of 9 are 1, 3, and 9 because 9 ÷ 1= 9
Step 2:
Count the factors:
The total number of factors
is
3 (1, 3, 9).
Since 9 has more than two factors, it is a composite number.
Check if 13 is composite:
Step 1:
Find factors of 13:
The factors of 13 are 1 and 13
Step 2:
Count the factors:
The total number of factors is 2 (1, 13).
Since 13 has exactly two factors, it is not composite (it is prime).
Check if 1 is composite:
Step 1:
Find factors of 1:
The only factor of 1 is 1 itself.
Since 1 has only one factor, it is neither prime nor composite.
Vertical Line
Types of Composite Number
Composite numbers are divided into two main types based on whether they are divisible by 2:
Odd Composite Numbers
These are odd numbers with more than two factors, meaning they are not prime. Examples include 9, 15, 21, 25, and 27. For instance:
9 has factors 1, 3, and 9.
15 has factors 1, 3, 5, and 15.
Odd composite numbers are not divisible by 2, but they can be divided evenly by other numbers besides 1 and themselves.
Even Composite Numbers
These are even numbers greater than 2 that have more than two factors. All even numbers except 2 (a prime number) are considered even composite. Examples include 4, 6, 8, 10, 12, and 14. For instance:
4 has factors 1, 2, and 4.
6 has factors 1, 2, 3, and 6.
Since these numbers are divisible by 2, they fall into the even composite category.
Properties of Composite Numbers
Composite numbers have several unique properties that can help students better understand their characteristics and applications:
More Than Two Factors:
Composite numbers always have more than two factors. This means that in addition to being divisible by 1 and themselves, they can also be divided evenly by at least one other positive integer. For example, 6 has factors 1, 2, 3, and 6, making it a composite number.
Smallest Composite Number:
Among all composite numbers, 4 is the smallest. It has three factors: 1, 2, and 4.
Divisibility by Smaller Numbers:
Composite numbers can always be divided evenly by smaller numbers. These smaller numbers can be either prime numbers (e.g., 2, 3) or other composite numbers (e.g., 4, 6). For example, 12 can be divided evenly by 2, 3, 4, and 6.
Special Relationship with 2
n
:
If n is a composite number, then 2
n
−1 is always a composite number. However, if n is a prime number, 2
n
−1 may or may not be prime. For instance:
When n = 4 (composite), 2
4
− 1 = 15, which is composite.
When n = 3 (prime), 2
3
− 1 = 7, which is prime.
Odd Number of Distinct Divisors in Perfect Squares:
Composite numbers that are perfect squares have an
odd number of distinct divisors
, unlike other numbers. For example:
9 (a perfect square) has divisors 1, 3, and 9—a total of three divisors.
4 (a perfect square) has divisors 1, 2, and 4—a total of three divisors. For all other composite numbers, divisors come in pairs, resulting in an even total number of divisors.
At Least Three Divisors:
Every composite number has at least three distinct divisors. For example, 4 has divisors 1, 2, and 4, while 8 has 1, 2, 4, and 8.
List of Composite Numbers from 1 to 100
|
4
|
20
|
33
|
46
|
58
|
72
|
85
|
96
|
|
6
|
21
|
34
|
48
|
60
|
74
|
86
|
98
|
|
8
|
22
|
35
|
49
|
62
|
75
|
87
|
99
|
|
9
|
24
|
36
|
50
|
63
|
76
|
88
|
100
|
|
10
|
25
|
38
|
51
|
64
|
77
|
90
|
|
|
12
|
26
|
39
|
52
|
65
|
78
|
91
|
|
|
14
|
27
|
40
|
54
|
66
|
80
|
92
|
|
|
15
|
28
|
42
|
55
|
68
|
81
|
93
|
|
|
16
|
30
|
44
|
56
|
69
|
82
|
94
|
|
|
18
|
32
|
45
|
57
|
70
|
84
|
95
|
|
Prime Factorization of Composite Numbers
Prime factorization is a method to express a composite number as a product of its prime factors. A prime number is one that has only two factors: 1 and itself. Composite numbers, on the other hand, have more than two factors and can be broken down into smaller prime numbers.
To perform prime factorization, divide the composite number by the smallest prime number (like 2, 3, 5, etc.) repeatedly until only prime numbers remain. Multiply these prime numbers together to get the original composite number.
For example:
Prime factorization of
24
:
S
tart dividing by 2
(the smallest prime number):
24 ÷ 2 = 12
12 ÷ 2 = 6
6 ÷ 2 = 3 (3 is prime, so stop here)
The prime factorization of 24 is 2 × 2 × 2 × 3
Prime factorization of
56
:
56 ÷ 2 = 28
28 ÷ 2 = 14
14 ÷ 2= 7 (7 is prime, so stop here.)
The prime factorization of 56 is 2 × 2 × 2 × 7
Prime factorization of
90
:
90 ÷ 2 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5 (5 is prime, so stop here)
The prime factorization of 90 is 2 × 3 × 3 × 5
Prime factorization of
120
:
120 ÷ 2 = 60
60 ÷ 2= 30
30 ÷ 2= 15
15 ÷ 3= 5 (5 is prime, so stop here)
The prime factorization of 120 is 2 × 2 × 2 × 3 × 5
Prime factorization of
210
210 ÷ 2 = 105
105 ÷ 3 = 35
35 ÷ 5 = 7 (7 is prime, so stop here)
The prime factorization of 210 is 2 × 3 × 5 × 7
Prime factorization of
360
:
360 ÷ 2 = 180
180 ÷ 2 = 90
90 ÷ 2 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5 (5 is prime, so stop here)
The prime factorization of 360 is 2× 2× 2× 3× 3× 5
This process shows how any composite number can be broken into its prime components, which are the "building blocks" of the number.
Interesting Facts About Composite Numbers
-
Every composite number can be written as a unique product of prime factors. For example, 12 = 2
2
× 3
-
The smallest composite number is 4, and there are infinitely many composite numbers.
-
Perfect squares, like 4 and 9, have an odd number of divisors, while other numbers have even divisors.
-
All even numbers except 2 are composite.
-
0 and 1 are neither composite nor prime.
-
The number of divisors of a composite number can be calculated using prime factorization. For example, 12 = 2
2
× 3 has 6 divisors: 1, 2, 3, 4, 6, 12.
These facts make composite numbers unique and essential in understanding number properties.
Composite Numbers Practice Questions
-
Which of the following are composite numbers: 11, 15, 23, 18.
-
How many composite numbers are there between 1 and 50?
-
Which is the smallest composite number, and why is it considered composite?
-
Write three examples each of odd composite numbers and even composite numbers.
-
Find the prime factorization of 24, 36, and 90
-
List all composite numbers between 10 and 30.
-
Fill in the missing composite numbers in this sequence: 4, __, 8, __, 12.
-
A teacher has 18 apples and wants to divide them equally among some students. Can the number of apples per student represent a composite number? Explain.
-
Find the product of the first four even composite numbers.
-
Find the sum of all composite numbers between 10 and 20.
