Question

Write the prime factorization of the integer.240

Short answer

The prime factorization of 240 is \(2^4 \times 3 \times 5\).

Step-by-step solution

Finding the first prime factor

Identify the smallest prime number that 240 can be divided by evenly. The smallest prime number is 2, and 240 is an even number, so it can be divided by 2. Performing the division \(240 ÷ 2\) results in 120.

Continuing the prime factorization

Continue this process of dividing by the smallest possible prime number. Now, divide 120 by 2, which results in 60. Repeat this process, dividing 60 by 2 to get 30, then divide 30 by 2 to get 15.

Prime factorization of remaining number

Now the result, 15, is not divisible by 2, so go to the next prime number which is 3. Divide 15 by 3 to get 5. Finally, divide 5 by 5 to get 1 which indicates the end of the process.

Key concepts

Prime Numbers

Prime numbers are the building blocks of whole numbers and are central to the concept of factorization. They are defined as numbers greater than 1 that have no divisors other than 1 and themselves. In other words, a prime number cannot be divided evenly by any other number except for 1 and the number itself.
For example, the prime numbers between 1 and 10 are 2, 3, 5, and 7. These numbers play a key role in factorization, as they are the 'atoms' that make up all other non-prime numbers through various combinations. The exercise of writing the prime factorization of the integer 240 is a practical application of understanding prime numbers.
Why are Prime Numbers Important for Prime Factorization?

• Unique Factorization: Every number can be written as a unique product of prime numbers, which is termed its prime factorization.
• Simplicity in Calculation: When breaking down numbers into their smallest components, using prime numbers simplifies calculations and analysis.
• Applications: Prime factorization has various applications in mathematics, computer science, and cryptography, due to the fundamental properties of prime numbers.

Integer Division

Integer division is a method of dividing two integers to produce another integer. In the context of factorization and particularly prime factorization, integer division is the process we use to simplify a number into its component factors by dividing it by prime numbers. When performing integer division, any remainder is discarded and only the quotient is taken into account.
For instance, in the provided exercise, 240 is divided by 2 to obtain a quotient of 120. This process is repeated with the result each time until the quotient is 1, which indicates that the number has been fully broken down into prime factors. Special attention should be paid that the division by any non-prime would not be as effective, as it could lead to a more complex factorization process.

Key Steps in Integer Division for Factorization:

• Begin with the smallest prime number.
• Divide the original number by this prime number to get a new quotient.
• Continue the process with the new quotient as the new number to be factored.
• Repeat the process using the next smallest prime number when the current prime number no longer divides the quotient evenly.

Factorization Methods

Factorization is the process of breaking down a composite number into its prime factors. There are several methods used to achieve this and prime factorization using integer division is a fundamental technique. But beyond this, understanding other factorization methods is also highly valuable.

Tree Diagram Method

The tree diagram method is a visual way of representing the factorization. It embraces a stepwise approach, dividing a number by prime numbers until only prime numbers are left, which then resemble the 'leaves' of the tree.

The Sieve of Eratosthenes

This ancient algorithm is used for finding all prime numbers up to any given limit. It systematically eliminates the multiples of prime numbers and the remaining numbers are primes that are used for factorization.

Usage of Factorization

The importance of factorization extends to solving mathematical problems including finding the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD). It's also pivotal for simplifying fractions and working with radicals. The ease of learning arithmetic or number theory is enhanced by a strong grasp of prime factorization and its related methods.