How Do You Factor a Number to Its Prime Factors?

To factor a number to its prime factors, you systematically divide the number by prime numbers until you are left with 1.

Here's a step-by-step breakdown:

1. Identify Prime Numbers: Remember that a prime number is a whole number greater than 1 that has only two factors: 1 and itself. Examples include 2, 3, 5, 7, 11, 13, and so on.

2. Start with the Smallest Prime Number (2): Check if the number you want to factor is divisible by 2 (i.e., if it's even).

3. Divide and Repeat:

   • If the number is divisible by 2, divide it by 2. Write down 2 as a prime factor.
   • Repeat step 3 with the result of the division until the result is no longer divisible by 2.

4. Move to the Next Prime Number (3): If the number is no longer divisible by 2, try dividing it by the next prime number, which is 3.

5. Continue the Process: Repeat the process of dividing by the next prime number (5, 7, 11, and so on) until the result of the division is 1.

6. List the Prime Factors: The prime factors are all the prime numbers you used to divide the original number.

Example: Factoring 36 to its Prime Factors

• 36 / 2 = 18 (2 is a prime factor)
• 18 / 2 = 9 (2 is a prime factor)
• 9 / 3 = 3 (3 is a prime factor)
• 3 / 3 = 1 (3 is a prime factor)

Therefore, the prime factorization of 36 is 2 x 2 x 3 x 3, or 2² x 3².

Another Example: Factoring 45 to its Prime Factors

• 45 is not divisible by 2.
• 45 / 3 = 15 (3 is a prime factor)
• 15 / 3 = 5 (3 is a prime factor)
• 5 / 5 = 1 (5 is a prime factor)

Therefore, the prime factorization of 45 is 3 x 3 x 5, or 3² x 5.

Why is Prime Factorization Important?

Prime factorization is a fundamental concept in number theory. It's useful in:

• Simplifying fractions: Finding the greatest common divisor (GCD).
• Finding the least common multiple (LCM): For example, when adding fractions with different denominators.
• Cryptography: Prime numbers play a vital role in encryption algorithms.