To decompose a number into its prime factors, you systematically divide the number by prime numbers until the quotient becomes 1. This process identifies the unique set of prime numbers that, when multiplied together, yield the original number.
Prime factorization is a fundamental concept in number theory, where 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.
The Simplest Algorithm for Prime Factorization
The most straightforward method to find the prime factors of a number involves successive division. As stated by BYJU'S, "The simplest algorithm to find the prime factors of a number is to keep on dividing the original number by prime factors until we get the remainder equal to 1."
Here’s a step-by-step breakdown of this method:
- Start with the smallest prime number: Begin by trying to divide the given number by the smallest prime number, which is 2.
- Divide repeatedly: If the number is divisible by 2, divide it and record 2 as a prime factor. Continue dividing the resulting quotient by 2 until it's no longer evenly divisible.
- Move to the next prime number: Once the number is no longer divisible by 2, try the next smallest prime number, which is 3. Repeat the division process.
- Continue with ascending prime numbers: Proceed to the next prime numbers (5, 7, 11, and so on) in ascending order, dividing the current quotient until it's no longer divisible by that prime.
- Stop when the quotient is 1: Keep going until the final quotient obtained from the division is 1. All the prime numbers you used as divisors are the prime factors of the original number.
Practical Example: Decomposing the Number 30
Let's illustrate this process with the number 30, as demonstrated in the BYJU'S reference:
| Step | Operation | Quotient | Prime Factor Found |
|---|---|---|---|
| 1 | 30 ÷ 2 | 15 | 2 |
| 2 | 15 ÷ 3 | 5 | 3 |
| 3 | 5 ÷ 5 | 1 | 5 |
Explanation:
- We start with 30. The smallest prime number is 2. 30 is divisible by 2, so we get 15. We record 2 as a prime factor.
- Now we have 15. Is 15 divisible by 2? No.
- The next prime number is 3. 15 is divisible by 3, so we get 5. We record 3 as a prime factor.
- Now we have 5. Is 5 divisible by 3? No.
- The next prime number is 5. 5 is divisible by 5, so we get 1. We record 5 as a prime factor.
- Since the quotient is 1, we stop.
Therefore, the prime factors of 30 are 2, 3, and 5. This means 30 can be expressed as the product:
30 = 2 × 3 × 5
This method ensures you find all prime factors efficiently by always testing the smallest possible prime divisors first.
Reference:
Definition, Methods, Examples, Prime Factorize - BYJU'S byjus.com
