Prime and Composite Numbers: Complete Notes | JNVS 2025

 

Prime and Composite Numbers: Complete Notes

Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In other words, a prime number can only be divided evenly by 1 and the number itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, and so on.

Properties of Prime Numbers

• Divisibility: A prime number is divisible only by 1 and itself.

• Uniqueness: Each prime number has exactly two factors: 1 and the number itself.

• No other divisors: A prime number cannot be factored into other smaller natural numbers, except for 1 and itself.

• The number 1 is not prime: 1 is neither a prime nor a composite number.

Examples of Prime Numbers

• 2 is the only even prime number. All other even numbers are divisible by 2, so they cannot be prime.

• 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc.

Why is 2 Special?

2 is the only even prime number. All other even numbers are divisible by 2 and thus are composite, not prime. This makes 2 the only even number in the prime category.

Prime Number Test

To check whether a number is prime, try dividing it by all integers less than or equal to the square root of the number. If none divide it evenly, then it is prime. For example, to check if 29 is prime:

• The square root of 29 is approximately 5.39, so check divisibility by 2, 3, and 5.

• Since 29 is not divisible by any of these, it is prime.

Applications of Prime Numbers

• Cryptography: Prime numbers are widely used in encryption algorithms, particularly in public key cryptography (e.g., RSA).

• Number Theory: Prime numbers form the building blocks for all numbers, as any number can be expressed as a product of primes (known as prime factorization).

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Composite Numbers

A composite number is a natural number greater than 1 that has more than two positive divisors. Unlike prime numbers, composite numbers can be divided evenly by numbers other than 1 and themselves.

Properties of Composite Numbers

• Divisibility: A composite number is divisible by numbers other than 1 and itself.

• Multiple Divisors: A composite number has more than two factors.

• Factorization: A composite number can be written as a product of two or more smaller natural numbers.

Examples of Composite Numbers

• 4 is a composite number because it can be factored as 2 × 2.

• 6 is a composite number because it can be factored as 2 × 3.

• 8 is a composite number because it can be factored as 2 × 2 × 2.

• 9 is a composite number because it can be factored as 3 × 3.

• 10 is a composite number because it can be factored as 2 × 5.

Why is 1 Not Composite?

1 is neither prime nor composite because it does not have two distinct divisors. The only divisor of 1 is itself.

Composite Numbers vs. Prime Numbers

• Prime numbers have exactly two factors: 1 and the number itself.

• Composite numbers have more than two factors.

Prime Numbers from 1 to 1000

2 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

101 to 200

• 101, 103, 107, 109, 113, 127, 131, 137, 139, 149

• 151, 157, 163, 167, 173, 179, 181, 191, 193, 197

• 199

201 to 300

• 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

• 269, 271, 277, 281, 283, 293

301 to 400

• 307, 311, 313, 317, 331, 337, 347, 349, 353, 359

• 367, 373, 379, 383, 389, 397

401 to 500

• 409, 419, 421, 431, 433, 439, 443, 449, 457, 461

• 463, 467, 479, 487, 491, 499

501 to 600

• 509, 521, 523, 541, 547, 557, 563, 569, 571, 577

• 587, 593, 599

601 to 700

• 607, 613, 617, 619, 631, 641, 643, 647, 653, 659

• 661, 673, 677, 683, 691, 701

701 to 800

• 709, 719, 727, 733, 739, 743, 751, 757, 761, 769

• 773, 787, 797

801 to 900

• 809, 811, 821, 823, 827, 829, 839, 853, 857, 859

• 863, 877, 881, 883, 887, 907

901 to 1000

• 911, 919, 929, 937, 941, 947, 953, 967, 971, 977

• 983, 991, 997

Prime Factorization of Composite Numbers

Every composite number can be expressed as a product of prime numbers. This process is called prime factorization. For example:

• 12 can be factorized as 2 × 2 × 3 (or 2² × 3).

• 30 can be factorized as 2 × 3 × 5.

Applications of Composite Numbers

• Factorization in Algebra: Composite numbers are used in factoring algebraic expressions.

• Mathematical Algorithms: Algorithms involving divisibility often utilize composite numbers.

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Key Differences Between Prime and Composite Numbers

Property	Prime Numbers	Composite Numbers
Definition	Numbers greater than 1 with exactly two factors (1 and itself).	Numbers greater than 1 with more than two factors.
Number of Factors	Exactly 2 factors.	More than 2 factors.
Example	2, 3, 5, 7, 11, 13, 17, 19	4, 6, 8, 9, 10, 12, 15, 18
Smallest Prime	2	No smallest composite number (starts from 4).
Divisibility	Only divisible by 1 and itself.	Divisible by numbers other than 1 and itself.
Prime Factorization	Cannot be factored further.	Can be expressed as a product of smaller prime numbers.
Even Prime	2 is the only even prime.	All other even numbers are composite.

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Practice Questions

Multiple Choice Questions (MCQs)

1. Which of the following is a prime number?

   • a) 15

   • b) 19

   • c) 25

   • d) 27

   • Answer: b) 19

2. Which of the following is a composite number?

   • a) 2

   • b) 3

   • c) 9

   • d) 11

   • Answer: c) 9

3. What is the prime factorization of 18?

   • a) 2 × 3 × 3

   • b) 2 × 2 × 3

   • c) 3 × 6

   • d) 2 × 9

   • Answer: a) 2 × 3 × 3

4. Which of the following numbers is neither prime nor composite?

   • a) 1

   • b) 2

   • c) 3

   • d) 4

   • Answer: a) 1

5. What is the smallest composite number?

   • a) 1

   • b) 2

   • c) 4

   • d) 6

   • Answer: c) 4

6. The number 29 is:

   • a) Composite

   • b) Prime

   • c) Neither prime nor composite

   • d) Cannot be determined

   • Answer: b) Prime

7. The number 45 has the following prime factorization:

   • a) 3 × 3 × 5

   • b) 2 × 3 × 3 × 5

   • c) 5 × 5 × 3

   • d) 3 × 15

   • Answer: a) 3 × 3 × 5

True/False Questions

1. 2 is the smallest prime number. (True/False)

   • Answer: True

2. All prime numbers are odd. (True/False)

   • Answer: False (2 is an even prime number)

3. 12 is a prime number. (True/False)

   • Answer: False (12 is a composite number)

4. Every composite number can be expressed as the product of prime numbers. (True/False)

   • Answer: True

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Conclusion

Understanding prime and composite numbers is fundamental to number theory and various branches of mathematics. Prime numbers serve as the building blocks of natural numbers, and composite numbers can be broken down into prime factors. Knowing how to identify, factorize, and differentiate these numbers is crucial for solving many mathematical problems, including those in exams like the JNV Entrance Test.

By practicing the prime factorization and solving questions, you can improve your understanding of these concepts and enhance your problem-solving skills in number theory.