Prime factorization involves breaking down a number into its prime number components. Here are some methods, tricks, and rules to aid in prime factorization:

Methods

1. Division Method:

   • Divide the number by the smallest prime number (usually 2) and continue dividing the quotient by prime numbers until the quotient is 1.
   • Example: Factorize 56
     • 56 ÷ 2 = 28
     • 28 ÷ 2 = 14
     • 14 ÷ 2 = 7
     • 7 ÷ 7 = 1
     • Prime factors: 2, 2, 2, 7 (or $2^3 \times 7$)

2. Factor Tree Method:

   • Draw a factor tree by breaking the number into two factors, then breaking those factors into smaller factors until all the factors are prime.
   • Example: Factorize 60
     • Start with 60 and break it down into 6 and 10.
     • Break 6 into 2 and 3, and 10 into 2 and 5.
     • Prime factors: 2, 2, 3, 5 (or $2^2 \times 3 \times 5$)

3. Using Prime Numbers List:

   • List prime numbers (2, 3, 5, 7, 11, etc.) and test each one as a factor of the number.
   • Example: Factorize 45
     • 45 ÷ 3 = 15
     • 15 ÷ 3 = 5
     • Prime factors: 3, 3, 5 (or $3^2 \times 5$)

Tricks

1. Divisibility Rules:

   • 2: A number is divisible by 2 if it is even.
   • 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
   • 5: A number is divisible by 5 if it ends in 0 or 5.
   • 7: Double the last digit and subtract it from the rest of the number; if the result is divisible by 7, so is the original number.
   • 11: Alternately subtract and add the digits of the number; if the result is divisible by 11, so is the original number.

2. Square Roots:

   • Only need to check divisibility up to the square root of the number.
   • Example: For 121, check divisibility up to $\sqrt{121} = 11$.

Rules

1. Uniqueness:

   • Every number has a unique prime factorization (Fundamental Theorem of Arithmetic).

2. Order:

   • Prime factors are usually written in ascending order.

3. Exponentiation:

   • Repeated prime factors can be expressed using exponents.
   • Example: $2^3 \times 3 \times 5$

Examples

• Example 1: Factorize 84

  • 84 ÷ 2 = 42
  • 42 ÷ 2 = 21
  • 21 ÷ 3 = 7
  • Prime factors: 2, 2, 3, 7 (or $2^2 \times 3 \times 7$)

• Example 2: Factorize 90

  • 90 ÷ 2 = 45
  • 45 ÷ 3 = 15
  • 15 ÷ 3 = 5
  • Prime factors: 2, 3, 3, 5 (or $2 \times 3^2 \times 5$)

Tip

Always start with the smallest prime number and work your way up. This ensures you find the prime factors systematically and avoid missing any factors.

Would you like more details or have any specific questions?

Here are 8 related questions to expand your understanding:

1. How do you determine if a large number is prime?
2. What is the prime factorization of 1001?
3. How can you quickly factorize numbers using mental math tricks?
4. How does the Euclidean algorithm help in finding the greatest common divisor (GCD) using prime factorization?
5. What is the significance of prime factorization in cryptography?
6. Can you explain how prime factorization is used to simplify fractions?
7. How would you find the least common multiple (LCM) using prime factors?
8. What are some real-world applications of prime factorization?

Tip: When factorizing large numbers, use a calculator or computer software to handle the division and checking for prime factors efficiently.