What are Prime and Composite Numbers?
Prime number: When we talk of prime numbers we define them to be a whole number greater than 1 with exactly two factors, 1 and itself. Each prime number is only divisible by 1 and itself.
For example, 2, 3, 5, 7, 11, 13, 17, 19, etc. are prime numbers.
Composite numbers: Composite numbers are a group of whole numbers that are greater than 1 with more than two factors.
For example, 4, 6, 8, 10, 30, 100, etc. are composite numbers. Each composite number can be divided by 1, by itself and at least one other number.
How to Find a Number that is Prime or Composite?
First, we need to find whether the number is even or odd. If the number is observed to be even and greater than 2, then it is considered to be a composite number.
First, we need to find whether the number is even or odd. If the number is observed to be even and greater than 2, then it is considered to be a composite number.
When a number is odd, we check for the following divisibility rules:
Divisibility rule of 3: If the sum of digits of a number is a multiple of 3, then, the number is divisible by 3.
Divisibility rule of 9: If the sum of the digits of a number is a multiple of 9, then, the number is divisible by 9.
Divisibility rule of 5: If the ones place digits of a number is 0 or 5, then, the number is divisible by 5.
For example,
Can you guess whether 56 is a prime or composite number?
56 is even, and greater than 2, hence it’s a composite number.
Is the number 21 a prime or composite number?
21 is an odd number. So, it is not divisible by 2 or any of its multiples.
Consider adding the two digits in 21, 2 + 1 = 3 . The sum, which is 3, is divisible by 3. So, 21 is divisible by 3.
Therefore, 21 has factors in addition to 1 and itself. This shows you that 21 is a composite number.
Factor Pairs
When two numbers are multiplied to get a number called the product, the two numbers form a factor pair of the product.
Let’s understand this concept with the help of finding the area of a rectangle.
We have a rectangle with an area of 16 square units. In the next step we need to find the number of rectangles of different dimensions that can be drawn with the same area of 16 square units?
The formula for the area of a rectangle = length × width
Therefore, 16 = 1 × 16, 16 = 2 × 8, 16 = 4 × 4
Each of these form what is known as a factor pair. 1 and 16, 2 and 8, 4 and 4, that is, the number or the product 16, can be formed by multiplying the factor pairs.
Now, If we take the area of a rectangle as 13 square units. Then, a possible rectangle can be drawn as,
13 = 1 × 13
There is only one possible rectangle. This tells us that the number 13 can only be formed by multiplying the only factor pair related to 13, which is i.e. 1 and 13. So 13 has only “1” factor pair, and hence it is a prime number.
Similarly, all prime numbers will have only 1 factor pair and composite numbers will have many or more than 1 factor pairs.
We can also use the factor pairs to identify primes and composites:
Numbers | Factor pairs | Prime or composite |
|---|---|---|
24 | ( 1,24 ), ( 2,12 ), ( 3,8 ), ( 4,6 ) | Composite |
21 | ( 1,21 ), ( 3,7 ) | Composite |
11 | ( 1,11 ) | Prime |
Solved Composite & Prime Number Examples
Example 1: Can you tell whether 47 is prime or composite?
Solution:
Use divisibility rule,
47 is odd, so it is not divisible by 2 or any other even number.
4 + 7 = 11. 11 is not divisible by 3 or 9. So, 47 is not divisible by 3 or 9.
The ones digit is not 0 or 5. So, 47 is not divisible by 5.
Therefore, 47 has exactly 2 factors, 1 and itself. So, 47 is a prime number.
Example 2:
There are 31 players in a tennis tournament. Can the coach divide these players into equal groups?
Solution:
To find the number of groups, we check the divisibility of number 31. If the number is divisible by any number, then we can divide the players into equal groups.
Now, by applying the divisibility rule:
31 is odd. So, it is not divisible by 2 or any other even number.
3 + 1 = 4. 4 is not divisible by 3 or 9. So, 31 is not divisible by 3 and 9.
The ones digit in 31 is not 0 or 5. So, it is not divisible by 5.
Therefore, 31 has exactly 2 factors, 1 and itself. So, 31 is a prime number, and the coach cannot divide the players into equal groups.
Example 3: Can you identify whether 11 is a prime or composite number?
Solution:
Use divisibility rule,
11 is an odd number. So, it is not divisible by 2 or any of its multiples.
1 + 1 = 2. 2 is not divisible by 3 or 9. So, 11 is not divisible by 3 or 9.
The ones digit is not 0 or 5. So, it is not divisible by 5.
11 has exactly two factors, 1 and itself.
Therefore, 11 is a prime number.
1 is neither a prime nor a composite number.
All even numbers except 2 are composite. 2 is the only number that is even and prime.
No odd number is divisible by an even number. Let’s take a few examples to understand, 25 is an odd number, it is not divisible by 2 or any of its multiples. Hence, 25 is not divisible by any even number. We can prove the same for any odd number. But some even numbers can be divisible by odd numbers for instance, 30 which is even is divisible by 3, which is an odd number.