Prime Factorization: Complete Notes for JNVS 2025

 

Prime Factorization: Complete Notes

1. Introduction to Prime Factorization

Prime factorization is the process of expressing a number as the product of prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself. When you factorize a number, you break it down into its prime factors. The prime factorization is unique for every number, except for the order of the factors (as per the Fundamental Theorem of Arithmetic).

For example, the prime factorization of 12 is:

$$12 = 2 \times 2 \times 3 \quad \text{or} \quad 12 = 2^2 \times 3$$

2. Steps to Perform Prime Factorization

1. Start with the number you want to factorize.

2. Divide by the smallest prime number (starting from 2) and check if it divides the number exactly (without remainder).

3. Repeat the process with the quotient obtained from the division until the quotient is a prime number.

4. Write down all the prime factors. If the number is divisible by a prime more than once, repeat the division.

5. Express the result as a product of prime factors.

3. Examples of Prime Factorization

Below are 20 examples demonstrating the prime factorization process.

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Prime Factorization Examples

1. 12:

   • 12 ÷ 2 = 6

   • 6 ÷ 2 = 3

   • 3 ÷ 3 = 1

   • Prime Factorization: $12 = 2^2 \times 3$

2. 15:

   • 15 ÷ 3 = 5

   • 5 ÷ 5 = 1

   • Prime Factorization: $15 = 3 \times 5$

3. 28:

   • 28 ÷ 2 = 14

   • 14 ÷ 2 = 7

   • 7 ÷ 7 = 1

   • Prime Factorization: $28 = 2^2 \times 7$

4. 30:

   • 30 ÷ 2 = 15

   • 15 ÷ 3 = 5

   • 5 ÷ 5 = 1

   • Prime Factorization: $30 = 2 \times 3 \times 5$

5. 45:

   • 45 ÷ 3 = 15

   • 15 ÷ 3 = 5

   • 5 ÷ 5 = 1

   • Prime Factorization: $45 = 3^2 \times 5$

6. 50:

   • 50 ÷ 2 = 25

   • 25 ÷ 5 = 5

   • 5 ÷ 5 = 1

   • Prime Factorization: $50 = 2 \times 5^2$

7. 60:

   • 60 ÷ 2 = 30

   • 30 ÷ 2 = 15

   • 15 ÷ 3 = 5

   • 5 ÷ 5 = 1

   • Prime Factorization: $60 = 2^2 \times 3 \times 5$

8. 72:

   • 72 ÷ 2 = 36

   • 36 ÷ 2 = 18

   • 18 ÷ 2 = 9

   • 9 ÷ 3 = 3

   • 3 ÷ 3 = 1

   • Prime Factorization: $72 = 2^3 \times 3^2$

9. 100:

   • 100 ÷ 2 = 50

   • 50 ÷ 2 = 25

   • 25 ÷ 5 = 5

   • 5 ÷ 5 = 1

   • Prime Factorization: $100 = 2^2 \times 5^2$

10. 120:

    • 120 ÷ 2 = 60

    • 60 ÷ 2 = 30

    • 30 ÷ 2 = 15

    • 15 ÷ 3 = 5

    • 5 ÷ 5 = 1

    • Prime Factorization: $120 = 2^3 \times 3 \times 5$

11. 135:

    • 135 ÷ 3 = 45

    • 45 ÷ 3 = 15

    • 15 ÷ 3 = 5

    • 5 ÷ 5 = 1

    • Prime Factorization: $135 = 3^3 \times 5$

12. 150:

    • 150 ÷ 2 = 75

    • 75 ÷ 3 = 25

    • 25 ÷ 5 = 5

    • 5 ÷ 5 = 1

    • Prime Factorization: $150 = 2 \times 3 \times 5^2$

13. 180:

    • 180 ÷ 2 = 90

    • 90 ÷ 2 = 45

    • 45 ÷ 3 = 15

    • 15 ÷ 3 = 5

    • 5 ÷ 5 = 1

    • Prime Factorization: $180 = 2^2 \times 3^2 \times 5$

14. 200:

    • 200 ÷ 2 = 100

    • 100 ÷ 2 = 50

    • 50 ÷ 2 = 25

    • 25 ÷ 5 = 5

    • 5 ÷ 5 = 1

    • Prime Factorization: $200 = 2^3 \times 5^2$

15. 225:

    • 225 ÷ 3 = 75

    • 75 ÷ 3 = 25

    • 25 ÷ 5 = 5

    • 5 ÷ 5 = 1

    • Prime Factorization: $225 = 3^2 \times 5^2$

16. 256:

    • 256 ÷ 2 = 128

    • 128 ÷ 2 = 64

    • 64 ÷ 2 = 32

    • 32 ÷ 2 = 16

    • 16 ÷ 2 = 8

    • 8 ÷ 2 = 4

    • 4 ÷ 2 = 2

    • 2 ÷ 2 = 1

    • Prime Factorization: $256 = 2^8$

17. 300:

    • 300 ÷ 2 = 150

    • 150 ÷ 2 = 75

    • 75 ÷ 3 = 25

    • 25 ÷ 5 = 5

    • 5 ÷ 5 = 1

    • Prime Factorization: $300 = 2^2 \times 3 \times 5^2$

18. 400:

    • 400 ÷ 2 = 200

    • 200 ÷ 2 = 100

    • 100 ÷ 2 = 50

    • 50 ÷ 2 = 25

    • 25 ÷ 5 = 5

    • 5 ÷ 5 = 1

    • Prime Factorization: $400 = 2^4 \times 5^2$

19. 450:

    • 450 ÷ 2 = 225

    • 225 ÷ 3 = 75

    • 75 ÷ 3 = 25

    • 25 ÷ 5 = 5

    • 5 ÷ 5 = 1

    • Prime Factorization: $450 = 2 \times 3^2 \times 5^2$

20. 500:

    • 500 ÷ 2 = 250

    • 250 ÷ 2 = 125

    • 125 ÷ 5 = 25

    • 25 ÷ 5 = 5

    • 5 ÷ 5 = 1

    • Prime Factorization: $500 = 2^2 \times 5^3$

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4. Applications of Prime Factorization

Prime factorization is used in many areas of mathematics and real-world applications:

• Finding the GCD (Greatest Common Divisor): Prime factorization helps to find the GCD of two numbers by identifying the common factors.

• Finding the LCM (Least Common Multiple): LCM can be found by using the prime factors of numbers.

• Simplifying Fractions: By canceling out common prime factors in the numerator and denominator.

• Cryptography: Many encryption algorithms, like RSA, are based on prime factorization.

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5. Practice Questions

Multiple Choice Questions (MCQs)

1. What is the prime factorization of 48?

   • a) $2^4 \times 3$

   • b) $2^3 \times 3^2$

   • c) $2^2 \times 3^3$

   • d) $2^5 \times 3$

   • Answer: a) $2^4 \times 3$

2. What is the prime factorization of 72?

   • a) $2^2 \times 3^3$

   • b) $2^3 \times 3^2$

   • c) $2^4 \times 3$

   • d) $2^5 \times 3^2$

   • Answer: b) $2^3 \times 3^2$

3. Which of the following is the correct prime factorization of 90?

   • a) $2 \times 3^2 \times 5$

   • b) $2^2 \times 3 \times 5$

   • c) $2 \times 3 \times 5^2$

   • d) $2 \times 3^3 \times 5$

   • Answer: a) $2 \times 3^2 \times 5$

4. What is the prime factorization of 54?

   • a) $2^3 \times 3$

   • b) $2^2 \times 3^3$

   • c) $2 \times 3^2$

   • d) $2^3 \times 5$

   • Answer: c) $2 \times 3^3$

5. Find the prime factorization of 144.

   • a) $2^2 \times 3^2$

   • b) $2^4 \times 3^2$

   • c) $2^3 \times 3^2$

   • d) $2^5 \times 3^2$

   • Answer: b) $2^4 \times 3^2$

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True/False Questions

1. The prime factorization of 100 is $2^2 \times 5^2$. (True/False)

   • Answer: True

2. The prime factorization of 16 is $2^4$. (True/False)

   • Answer: True

3. The prime factorization of 36 is $2^2 \times 3^2$. (True/False)

   • Answer: True

4. The prime factorization of 200 is $2^2 \times 5^3$. (True/False)

   • Answer: True

5. The prime factorization of 25 is $2^2 \times 5$. (True/False)

   • Answer: False (Correct answer: $5^2$)

Alright!
I’ll prepare complete notes on Prime Factorisation for you — explaining the concept in detail, providing 20 worked examples and practice exercises. The total will be around 1000 words so it’s comprehensive and exam-ready.

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Prime Factorisation – Complete Notes

1. Introduction

Prime factorisation is the process of expressing a number as the product of its prime numbers.
A prime number is a number greater than 1 that has exactly two factors: 1 and itself.
Examples of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, …

Numbers that are not prime and can be expressed as a product of primes are called composite numbers.
Examples: 4 (2 × 2), 12 (2 × 2 × 3), 20 (2 × 2 × 5).

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2. Why Prime Factorisation is Important

• It helps in finding the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of numbers.

• It is useful in simplifying fractions.

• It’s a basic tool in number theory.

• It’s used in cryptography (internet security).

• It’s needed in algebra and problem-solving.

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3. Methods of Prime Factorisation

Method 1: Factor Tree Method

1. Write the number.

2. Break it into any two factors.

3. Continue breaking composite factors until only prime numbers remain.

4. Multiply all prime numbers to verify.

Example: 36
36 → 6 × 6
6 → 2 × 3
6 → 2 × 3
Prime factors: 2 × 2 × 3 × 3 or $2^2 × 3^2$

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Method 2: Division Method

1. Start dividing the number by the smallest prime number (2).

2. Continue dividing until you cannot divide further.

3. Move to the next prime number (3, 5, 7…).

4. Stop when the quotient becomes 1.

Example: 84
84 ÷ 2 = 42
42 ÷ 2 = 21
21 ÷ 3 = 7
7 ÷ 7 = 1
Prime factors: 2 × 2 × 3 × 7 or $2^2 × 3 × 7$

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4. Prime Factorisation of First 20 Numbers

Number	Prime Factorisation
2	2
3	3
4	2 × 2 = $2^2$
5	5
6	2 × 3
7	7
8	2 × 2 × 2 = $2^3$
9	3 × 3 = $3^2$
10	2 × 5
12	2 × 2 × 3 = $2^2 × 3$
14	2 × 7
15	3 × 5
16	2 × 2 × 2 × 2 = $2^4$
18	2 × 3 × 3 = $2 × 3^2$
20	2 × 2 × 5 = $2^2 × 5$
21	3 × 7
22	2 × 11
24	2 × 2 × 2 × 3 = $2^3 × 3$
28	2 × 2 × 7 = $2^2 × 7$
30	2 × 3 × 5

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5. 20 Worked Examples

Example 1: 45
45 ÷ 3 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1
Prime factors: $3^2 × 5$

Example 2: 64
64 ÷ 2 = 32
32 ÷ 2 = 16
16 ÷ 2 = 8
8 ÷ 2 = 4
4 ÷ 2 = 2
2 ÷ 2 = 1
Prime factors: $2^6$

Example 3: 72
72 ÷ 2 = 36
36 ÷ 2 = 18
18 ÷ 2 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
Prime factors: $2^3 × 3^2$

Example 4: 90 → $2 × 3^2 × 5$
Example 5: 100 → $2^2 × 5^2$
Example 6: 105 → $3 × 5 × 7$
Example 7: 108 → $2^2 × 3^3$
Example 8: 120 → $2^3 × 3 × 5$
Example 9: 150 → $2 × 3 × 5^2$
Example 10: 175 → $5^2 × 7$
Example 11: 180 → $2^2 × 3^2 × 5$
Example 12: 196 → $2^2 × 7^2$
Example 13: 200 → $2^3 × 5^2$
Example 14: 225 → $3^2 × 5^2$
Example 15: 256 → $2^8$
Example 16: 300 → $2^2 × 3 × 5^2$
Example 17: 360 → $2^3 × 3^2 × 5$
Example 18: 400 → $2^4 × 5^2$
Example 19: 420 → $2^2 × 3 × 5 × 7$
Example 20: 500 → $2^2 × 5^3$

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6. Practice Exercises

Find the prime factorisation of the following numbers:

1. 84

2. 96

3. 144

4. 150

5. 168

6. 180

7. 196

8. 210

9. 225

10. 240

11. 256

12. 288

13. 300

14. 320

15. 360

16. 375

17. 400

18. 420

19. 450

20. 500

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7. Tips to Remember

• 2 is the only even prime number.

• If a number ends in 0, 2, 4, 6, 8 → divisible by 2.

• If the sum of digits is divisible by 3 → divisible by 3.

• If a number ends in 0 or 5 → divisible by 5.

• Large numbers can be broken into smaller factors to make factorisation easier.

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8. Summary

Prime factorisation is an essential concept in mathematics.
It involves breaking down a number into the product of prime numbers, using either:

• Factor Tree Method, or

• Division Method.

It is a foundation for advanced math concepts like HCF, LCM, and algebraic problem solving.

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If you want, I can also prepare this as a PDF with diagrams (factor trees) so it’s visually easy for students to follow.
Do you want me to make it into a PDF with colourful factor tree charts? That would make it perfect for study material.

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Conclusion

Prime factorization is a powerful tool in mathematics, helping solve problems in number theory, algebra, and more. By practicing the prime factorization of numbers and understanding its applications, you will be well-prepared for the JNV Entrance Test 2025. Regular practice with these examples will enhance your skills in breaking down numbers into their prime factors efficiently.