What is Prime Factorization?

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Discover What is Prime Factorization? Here and discover the secrets of numbers. Learn how to break down any integer into its unique prime factors with this comprehensive guide.

What is Prime Factorization?

Prime factorization is a mathematical process used to express a positive integer as a product of its prime factors. In other words, it breaks down a number into its smallest possible prime factors.

To find the prime factorization of a number, you divide the number by its smallest prime factor and keep dividing the resulting quotient by its smallest prime factor until the quotient becomes 1. The prime factors are the prime numbers that are used in the divisions.

For example, let’s find the prime factorization of the number 36:

• Start with the number 36.
• The smallest prime factor of 36 is 2. Divide 36 by 2 to get 18.
• The smallest prime factor of 18 is 2. Divide 18 by 2 to get 9.
• The smallest prime factor of 9 is 3. Divide 9 by 3 to get 3.
• The smallest prime factor of 3 is 3. Divide 3 by 3 to get 1.
• The prime factors of 36 are 2, 2, 3 and 3. Written as a product, the prime factorization of 36 is 2^2 × 3^2 (where “^” indicates exponentiation).

Prime factorization is useful in various areas of mathematics, including number theory, cryptography, and solving certain types of equations. It helps to understand the properties and behavior of numbers, and it is also used to simplify fractions, find the greatest common divisor (GCD), and determine the least common multiple (LCM) of numbers.

How to Find Prime Factorization?

Finding the prime factorization of a number involves expressing that number as a product of its prime factors. To find the prime factorization, you can follow these steps:

• Start with the number you want to factor.
• Start by dividing the number by the smallest prime number, which is 2. Continue dividing until the number is no longer divisible by 2.
• Move on to the next prime, which is 3, and repeat the division process until the number is no longer divisible by 3.
• Continue this process with successive primes (5, 7, 11, 13 and so on) until you have exhausted all the primes up to the square root of the original number. It is not necessary to test divisibility by primes greater than the square root because if there are any factors greater than the square root, there must also be factors smaller than the square root.
• The remaining number that cannot be divided further is the largest prime factor of the original number.
• Write all the prime factors obtained by the divisions.

Let’s go through an example:

Example: Find the prime factorization of 84.

Start with the number 84.

Divide by 2: 84 ÷ 2 = 42. Keep dividing by 2 until it is no longer divisible: 42 ÷ 2 = 21.

Divide by 3: 21 ÷ 3 = 7. Now we have a prime factor (3).

Since 7 is prime, we have obtained all the prime factors.

Write the prime factors: 2 × 2 × 3 × 7 = 84.

So, the prime factorization of 84 is 2 × 2 × 3 × 7.

Remember, this method works for any number, but larger numbers may require more iterations and possibly larger prime numbers.

How to Find Prime Factors of a Large Number?

Finding the prime factors of a large number can be a complex task, but there are several methods you can use to tackle it. Here is a step-by-step approach using a combination of trial division and the Sieve of Eratosthenes method:

Start with the smallest prime, which is 2, and check if it divides the given number evenly. If so, divide the number by 2 and keep dividing until it is no longer divisible by 2. Keep track of the number of times you divide by 2.

Move to the next prime, which is 3, and check if it divides the remaining number evenly. If so, divide the number by 3 and continue dividing until it is no longer divisible by 3. Again, keep track of the number of times you divide by 3.

Continue this process with each subsequent prime, checking to see if it divides the remaining number evenly and dividing until it is no longer divisible. The primes to consider are: 2, 3, 5, 7, 11, 13, 17, 19, 23, etc.

Repeat the above steps until the remaining number becomes 1. At this point, you have found all the prime factors of the original large number.

Let’s illustrate this method with an example:

Example: Find the prime factors of 840.

Step 1: Divide 840 by 2 until it is no longer divisible: 840 ÷ 2 = 420.

Step 2: Divide 420 by 2 until it is no longer divisible: 420 ÷ 2 = 210.

Step 3: Divide 210 by 2 until it is no longer divisible: 210 ÷ 2 = 105.

Step 4: Now, 105 is not divisible by 2, so move to the next prime which is 3.

Divide 105 by 3 until it is no longer divisible: 105 ÷ 3 = 35.

Step 5: Divide 35 by 5 until it is no longer divisible: 35 ÷ 5 = 7.

Step 6: 7 is prime itself, so the process ends here.

The prime factors of 840 are 2, 2, 2, 3, 5 and 7.

Note that if the given number is extremely large, trial division alone may not be efficient enough. In such cases, more advanced algorithms, such as Pollard’s rho algorithm or the quadratic sieve method, can be used to find the prime factors. These methods are beyond the scope of this explanation, but they are worth investigating if you need to factor extremely large numbers.

Prime Factorization of HCF and LCM

The prime factorization of the highest common factor (HCF) and the least common multiple (LCM) can be determined using the prime factorization method. Here’s how you can find the prime factorization of the HCF and LCM:

Prime factorization of HCF:

Start by finding the prime factors of all the numbers you want to find the HCF of.

Identify the common prime factors among all the numbers.

Multiply these common prime factors together to get the HCF.

Prime factorization of LCM:

Start by finding the prime factors of all the numbers you want to find the LCM of.

Combine all distinct prime factors of all numbers.

Multiply these separate prime factors together to get the LCM.

Let’s take an example to illustrate this process:

Example: Find the prime factorization of the HCF and LCM of 12, 18, and 24.

Prime factorization of HCF:

• Prime factors of 12: 2^2 * 3^1
• Prime factors of 18: 2^1 * 3^2
• Prime factors of 24: 2^3 * 3^1
• The common prime factors are 2^1 * 3^1, so the HCF = 2^1 * 3^1 = 6.

Prime factorization of LCM:

Combine all the distinct prime factors: 2^3 * 3^2 = 72.

The MCM = 72.

Therefore, the prime factorization of the HCF is 2^1 * 3^1 = 6, and the prime factorization of the LCM is 2^3 * 3^2 = 72.

Prime Factoring Solved Examples

Here are some solved examples of prime factorization:

Example 1: Find the prime factorization of 56.

To determine the prime factorization of 56, we divide the number by its smallest prime factor and continue the process until we reach 1.

• Dividing 56 by 2 gives us 28.
• Dividing 28 by 2 gives us 14.
• Dividing 14 by 2 gives us 7.
• Since 7 is prime, we have completed the prime factorization.

Therefore, the prime factorization of 56 is 2 * 2 * 2 * 7, or simply written as 2^3 * 7.

Example 2: Find the prime factorization of 90.

• Dividing 90 by 2 gives us 45.
• Dividing 45 by 3 gives us 15.
• Dividing 15 by 3 gives us 5.
• Since 5 is prime, we have completed the prime factorization.

Therefore, the prime factorization of 90 is 2 * 3 * 3 * 5, or simply written as 2 * 3^2 * 5.

Example 3: Find the prime factorization of 120.

• Dividing 120 by 2 gives us 60.
• Dividing 60 by 2 gives us 30.
• Dividing 30 by 2 gives us 15.
• Dividing 15 by 3 gives us 5.

Since 5 is prime, we have completed the prime factorization.

Therefore, the prime factorization of 120 is 2 * 2 * 2 * 3 * 5, or simply written as 2^3 * 3 * 5.

Example 4: Find the prime factorization of 144.

• Dividing 144 by 2 gives us 72.
• Dividing 72 by 2 gives us 36.
• Dividing 36 by 2 gives us 18.
• Dividing 18 by 2 gives us 9.
• Dividing 9 by 3 gives us 3.

Since 3 is prime, we have completed the prime factorization.

Therefore, the prime factorization of 144 is 2 * 2 * 2 * 2 * 3 * 3, or simply written as 2^4 * 3^2.

These examples show how to find the prime factorization of a number by repeatedly dividing it by prime factors until only primes remain.

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