Prime and Composite Numbers
Let us play a game with numbers. Take the number 7. Try to divide it by 2, 3, 4, 5, or 6. None of them divide 7 exactly. The only numbers that divide 7 are 1 and 7 itself. Now take the number 12. You can divide 12 by 2, 3, 4, and 6. So 12 has many divisors. Numbers like 7, which have only two factors (1 and itself), are called prime numbers. Numbers like 12, which have more than two factors, are called composite numbers. Prime numbers are like the building blocks of all numbers. Just like every building is made of bricks, every number can be built by multiplying prime numbers together. This idea is so important that it has a special name: the Fundamental Theorem of Arithmetic. In this chapter, we will learn to identify prime and composite numbers, discover the Sieve of Eratosthenes (a clever ancient method to find primes), and understand why prime numbers are so special. This is part of the Playing with Numbers chapter in Grade 6 Maths.
What is Prime and Composite Numbers - Grade 6 Maths (Playing with Numbers)?
Prime Number: A number greater than 1 that has exactly two factors: 1 and the number itself. In other words, a prime number cannot be divided evenly by any number other than 1 and itself.
Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ...
Composite Number: A number greater than 1 that has more than two factors. In other words, a composite number can be divided evenly by at least one number other than 1 and itself.
Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, ...
What about 1? The number 1 is neither prime nor composite. It has only one factor (itself), but prime numbers must have exactly two factors. So 1 is in a special category all by itself.
What about 0? Zero is also neither prime nor composite. Zero can be divided by any non-zero number, so it has infinitely many factors. Prime and composite definitions only apply to numbers greater than 1.
Key fact: 2 is the only even prime number. Every other even number is divisible by 2 (and also by 1 and itself), so they have at least 3 factors and are composite. All prime numbers greater than 2 are odd.
Co-prime numbers: Two numbers are co-prime if they have no common factor other than 1. For example, 8 and 15 are co-prime because their only common factor is 1. Note: co-prime numbers do not have to be prime themselves.
Prime and Composite Numbers Formula
There is no formula to generate prime numbers, but there are useful facts and methods:
Checking if a number is prime:
To check if a number N is prime, try dividing it by all prime numbers up to the square root of N. If none of them divide N exactly, then N is prime.
For example, to check if 29 is prime: The square root of 29 is about 5.4. So check primes up to 5: 2, 3, 5.
29 / 2 = 14 remainder 1 (not divisible)
29 / 3 = 9 remainder 2 (not divisible)
29 / 5 = 5 remainder 4 (not divisible)
Since no prime up to 5 divides 29, it is prime.
Prime numbers up to 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
There are 25 prime numbers from 1 to 100.
Sieve of Eratosthenes: A method to find all primes up to a given number. Write all numbers from 2 to N. Start with 2 (the first prime). Cross out all multiples of 2 (except 2 itself). Move to the next uncrossed number (3). Cross out all multiples of 3. Continue with 5, 7, etc. The remaining uncrossed numbers are all prime.
Prime Factorisation: Every composite number can be written as a product of prime numbers. For example, 60 = 2 x 2 x 3 x 5. This is called the prime factorisation of 60.
Derivation and Proof
Let us use the Sieve of Eratosthenes to find all prime numbers up to 50. This is an ancient method invented by a Greek mathematician named Eratosthenes about 2,200 years ago.
Step 1: Write all numbers from 2 to 50:
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
Step 2: Circle 2 (it is prime). Cross out all multiples of 2 greater than 2: 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50.
Step 3: The next uncrossed number is 3. Circle it. Cross out all multiples of 3 that are not already crossed: 9, 15, 21, 27, 33, 39, 45.
Step 4: The next uncrossed number is 5. Circle it. Cross out multiples of 5 not already crossed: 25, 35.
Step 5: The next uncrossed number is 7. Circle it. Cross out multiples of 7 not already crossed: 49.
Step 6: The next uncrossed number is 11. Since 11 x 11 = 121 > 50, we can stop. All remaining uncrossed numbers are prime.
Prime numbers up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
There are 15 prime numbers between 1 and 50.
The sieve works because every composite number has a prime factor less than or equal to its square root. By crossing out multiples of each prime, we eliminate all composite numbers, leaving only primes.
Types and Properties
Here are the types of problems you will encounter:
Type 1: Identifying Prime and Composite Numbers - Given a number, determine whether it is prime, composite, or neither (in the case of 1). Check its factors to decide.
Type 2: Listing Primes in a Range - List all prime numbers in a given range, like between 30 and 50. Use the Sieve or direct testing.
Type 3: Sieve of Eratosthenes - Apply the sieve method to find all primes up to a given number. This is a step-by-step elimination process.
Type 4: Prime Factorisation - Break a composite number into a product of prime factors. Use the factor tree method or repeated division method.
Type 5: Twin Primes - Find pairs of prime numbers that differ by 2. Examples: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43).
Type 6: Co-prime Numbers - Determine if two numbers are co-prime (have no common factor other than 1). For example, are 14 and 15 co-prime? Factors of 14: 1,2,7,14. Factors of 15: 1,3,5,15. Common factor: only 1. Yes, co-prime.
Type 7: True/False and Reasoning - Questions like "Is every prime number odd?" (No, 2 is even and prime), "Is every odd number prime?" (No, 9 is odd but composite), or "Is 1 prime?" (No).
Solved Examples
Example 1: Example 1: Is 37 Prime or Composite?
Problem: Determine if 37 is prime or composite.
Solution:
To check, divide 37 by all primes up to the square root of 37. The square root of 37 is about 6.1. So we check primes up to 6: 2, 3, 5.
37 / 2 = 18 remainder 1 (not divisible by 2)
37 / 3 = 12 remainder 1 (not divisible by 3)
37 / 5 = 7 remainder 2 (not divisible by 5)
Since no prime up to 6 divides 37 exactly, 37 is a prime number.
Example 2: Example 2: Is 51 Prime or Composite?
Problem: Determine if 51 is prime or composite.
Solution:
Square root of 51 is about 7.1. Check primes up to 7: 2, 3, 5, 7.
51 / 2 = 25 remainder 1 (not divisible)
51 / 3 = 17 (exactly divisible!)
Since 3 divides 51 exactly, 51 has factors other than 1 and itself (1, 3, 17, 51).
51 is a composite number. 51 = 3 x 17.
Example 3: Example 3: Listing Primes Between 20 and 40
Problem: List all prime numbers between 20 and 40.
Solution:
Check each number:
21 = 3 x 7 (composite), 22 = 2 x 11 (composite), 23 (prime - not divisible by 2, 3, or any prime up to 4.8), 24 = 2 x 12 (composite), 25 = 5 x 5 (composite), 26 = 2 x 13 (composite), 27 = 3 x 9 (composite), 28 = 4 x 7 (composite), 29 (prime), 30 = 2 x 15 (composite), 31 (prime), 32 = 2 x 16 (composite), 33 = 3 x 11 (composite), 34 = 2 x 17 (composite), 35 = 5 x 7 (composite), 36 = 6 x 6 (composite), 37 (prime), 38 = 2 x 19 (composite), 39 = 3 x 13 (composite).
Prime numbers between 20 and 40: 23, 29, 31, 37
Example 4: Example 4: Prime Factorisation Using Factor Tree
Problem: Find the prime factorisation of 72.
Solution using Factor Tree:
72 = 2 x 36
36 = 2 x 18
18 = 2 x 9
9 = 3 x 3
So: 72 = 2 x 2 x 2 x 3 x 3
Prime factorisation of 72 = 2 x 2 x 2 x 3 x 3 = 2 cubed x 3 squared
Example 5: Example 5: Prime Factorisation Using Division Method
Problem: Find the prime factorisation of 180.
Solution:
Divide by the smallest prime factor repeatedly:
180 / 2 = 90
90 / 2 = 45
45 / 3 = 15
15 / 3 = 5
5 / 5 = 1
Prime factorisation of 180 = 2 x 2 x 3 x 3 x 5
Example 6: Example 6: Finding Twin Primes
Problem: Find all twin prime pairs between 1 and 50.
Solution:
Twin primes are pairs of prime numbers that differ by exactly 2.
Primes up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Check pairs with a difference of 2:
(3, 5): 5 - 3 = 2. Twin primes!
(5, 7): 7 - 5 = 2. Twin primes!
(11, 13): 13 - 11 = 2. Twin primes!
(17, 19): 19 - 17 = 2. Twin primes!
(29, 31): 31 - 29 = 2. Twin primes!
(41, 43): 43 - 41 = 2. Twin primes!
Twin prime pairs between 1 and 50: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43)
Example 7: Example 7: Checking Co-prime Numbers
Problem: Are 15 and 28 co-prime?
Solution:
Factors of 15: 1, 3, 5, 15
Factors of 28: 1, 2, 4, 7, 14, 28
Common factors: only 1
Yes, 15 and 28 are co-prime because their only common factor is 1.
Note: Neither 15 nor 28 is a prime number, but they can still be co-prime.
Example 8: Example 8: Express as Sum of Two Primes
Problem: Express 36 as the sum of two prime numbers in different ways.
Solution:
We need to find two primes that add up to 36:
36 = 5 + 31 (both prime)
36 = 7 + 29 (both prime)
36 = 13 + 23 (both prime)
36 = 17 + 19 (both prime)
36 can be expressed as the sum of two primes in 4 ways.
(This is related to Goldbach's Conjecture, which says every even number greater than 2 can be written as the sum of two primes.)
Example 9: Example 9: True or False with Reasoning
Problem: True or False: "All prime numbers are odd."
Solution:
False. The number 2 is a prime number and it is even. 2 is divisible only by 1 and 2, so it has exactly two factors, making it prime.
However, 2 is the ONLY even prime number. Every other even number is divisible by 2, so they have at least three factors (1, 2, and the number itself) and are composite.
Example 10: Example 10: The Number 1
Problem: Why is 1 neither prime nor composite?
Solution:
A prime number must have exactly 2 factors (1 and itself).
A composite number must have more than 2 factors.
The number 1 has only 1 factor: itself (1). Since 1 has exactly 1 factor (not 2 and not more than 2), it is neither prime nor composite.
This is important because if 1 were considered prime, then the prime factorisation of a number would not be unique (you could always multiply by 1 any number of times).
Real-World Applications
Prime numbers have many important applications. In your daily life, you use prime factorisation when simplifying fractions, finding HCF and LCM, and solving division problems.
In computer science, prime numbers are the foundation of internet security. When you shop online or send a message through WhatsApp, your data is encrypted using very large prime numbers. The security works because it is extremely difficult to find the prime factors of very large numbers (with hundreds of digits). This is called RSA encryption, and it keeps your passwords and bank details safe.
In nature, some interesting patterns involve primes. Cicadas (a type of insect) emerge from underground every 13 or 17 years, both prime numbers. Scientists believe this helps them avoid predators that appear in cycles of 2, 3, 4, or 6 years, because prime numbers have no common factors with these cycles.
Prime factorisation is used in engineering for gear design. When designing gears, engineers use prime numbers to avoid patterns that cause wear on the same teeth repeatedly.
In mathematics, prime numbers are considered the atoms of the number world. Just as every substance is made of atoms, every whole number greater than 1 is either a prime or can be built by multiplying primes. Understanding primes means understanding the fundamental structure of numbers.
Key Points to Remember
- A prime number has exactly 2 factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13, ...
- A composite number has more than 2 factors. Examples: 4, 6, 8, 9, 10, 12, ...
- 1 is neither prime nor composite (it has only 1 factor).
- 2 is the smallest prime number and the only even prime number.
- To check if a number is prime, divide it by all primes up to its square root.
- The Sieve of Eratosthenes is a method to find all primes up to a given number by crossing out multiples.
- There are 25 prime numbers between 1 and 100.
- Every composite number can be expressed as a product of prime numbers (prime factorisation).
- Twin primes are pairs of primes that differ by 2, like (3,5), (11,13), (17,19).
- Two numbers are co-prime if their only common factor is 1. Co-prime numbers need not be prime themselves.
Practice Problems
- Classify each number as prime, composite, or neither: 1, 2, 9, 17, 23, 27, 31, 49.
- List all prime numbers between 50 and 70.
- Use the Sieve of Eratosthenes to find all primes up to 30.
- Find the prime factorisation of 120 using the factor tree method.
- Find the prime factorisation of 225 using the division method.
- Find all twin prime pairs between 50 and 100.
- Are 16 and 25 co-prime? Justify your answer.
- Express 50 as the sum of two prime numbers.
Frequently Asked Questions
Q1. Why is 1 not a prime number?
A prime number must have exactly 2 distinct factors. The number 1 has only 1 factor (itself). If 1 were called prime, then the Fundamental Theorem of Arithmetic (which says every number has a unique prime factorisation) would break, because you could write 12 = 2 x 2 x 3 or 12 = 1 x 2 x 2 x 3 or 12 = 1 x 1 x 2 x 2 x 3, all different. To keep prime factorisation unique, 1 is excluded from primes.
Q2. Is 2 the only even prime number?
Yes. 2 is the only even prime number. Every other even number (4, 6, 8, 10, ...) is divisible by 2, so they have at least 3 factors (1, 2, and themselves), making them composite. 2 is special because its only factors are 1 and 2.
Q3. Are there infinitely many prime numbers?
Yes. The ancient Greek mathematician Euclid proved over 2,000 years ago that there are infinitely many primes. No matter how far you go in the number line, you will always find more prime numbers. They become less frequent as numbers get larger, but they never stop appearing.
Q4. What is the largest known prime number?
As of now, the largest known prime number has millions of digits. These massive primes are found using computers. They are called Mersenne primes (primes of the form 2 to the power p minus 1). The search for bigger primes is still going on. For your exam, you only need to know primes up to 100.
Q5. What is the difference between prime numbers and co-prime numbers?
A prime number is a single number with exactly 2 factors. Co-prime is a relationship between two numbers whose only common factor is 1. Co-prime numbers do not have to be prime. For example, 8 and 15 are co-prime (common factor is only 1), but neither 8 nor 15 is a prime number.
Q6. How is the Sieve of Eratosthenes useful?
It is the fastest manual method to find all prime numbers up to a given limit. Instead of checking each number individually, you systematically eliminate composites by crossing out multiples. For finding all primes up to 100, it is much faster than testing each number separately.
Q7. Can a prime number end in 0, 2, 4, 5, 6, or 8?
Except for 2 and 5, no prime number ends in 0, 2, 4, 5, 6, or 8. Numbers ending in 0, 2, 4, 6, 8 are even (divisible by 2). Numbers ending in 0 or 5 are divisible by 5. So any prime greater than 5 must end in 1, 3, 7, or 9. But not all numbers ending in these digits are prime (e.g., 21 = 3 x 7).
Q8. What is prime factorisation?
Prime factorisation is expressing a number as a product of prime numbers. Every composite number has a unique prime factorisation. For example, 60 = 2 x 2 x 3 x 5. No matter how you factorise 60, you will always end up with the same set of prime factors. This uniqueness is guaranteed by the Fundamental Theorem of Arithmetic.
Q9. Is there a formula to generate all prime numbers?
No, there is no simple formula that generates all prime numbers. This is one of the great mysteries of mathematics. Primes follow no regular pattern. We have methods to test if a number is prime, and sieves to find primes in a range, but no formula that gives the nth prime directly.
Q10. Why are prime numbers important in real life?
Prime numbers are crucial for internet security (encryption). Every time you make an online payment, send a private message, or log into a website, prime numbers protect your data. They are also used in coding theory, hash functions in computer science, and even in nature (some insect life cycles follow prime number patterns to avoid predators).
