The prime factorization of 384 is 2^7 × 3.
Understanding Prime Factorization
Prime factorization is the process of breaking down a composite number into its prime factors. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11). Every composite number can be expressed as a unique product of prime numbers. This is known as the Fundamental Theorem of Arithmetic.
How to Find Prime Factorization
The most common methods for finding prime factorization include:
- Factor Tree Method: Start by dividing the number by the smallest prime number that divides it evenly. Continue dividing the quotients by prime numbers until all branches end in prime numbers.
- Division Method: Repeatedly divide the number by the smallest possible prime number until the quotient is 1.
Prime Factorization of 384: A Step-by-Step Breakdown
Let's find the prime factorization of 384 using the division method:
- Start with 384. It is an even number, so divide by 2:
- 384 ÷ 2 = 192
- 192 is even, divide by 2:
- 192 ÷ 2 = 96
- 96 is even, divide by 2:
- 96 ÷ 2 = 48
- 48 is even, divide by 2:
- 48 ÷ 2 = 24
- 24 is even, divide by 2:
- 24 ÷ 2 = 12
- 12 is even, divide by 2:
- 12 ÷ 2 = 6
- 6 is even, divide by 2:
- 6 ÷ 2 = 3
- 3 is a prime number, divide by 3:
- 3 ÷ 3 = 1
Collecting all the prime divisors, we get:
2 × 2 × 2 × 2 × 2 × 2 × 2 × 3
This can be written in exponential form as 2^7 × 3.
Prime Factors and Factor Pairs of 384
The prime factors of 384 are 2 and 3.
The distinct factor pairs of 384 are combinations of two numbers that multiply together to give 384. These include:
| Factor 1 | Factor 2 |
|---|---|
| 1 | 384 |
| 2 | 192 |
| 3 | 128 |
| 4 | 96 |
| 6 | 64 |
| 8 | 48 |
| 12 | 32 |
| 16 | 24 |
