Question

Write the prime factorization of the integer.150

Short answer

The prime factorization of 150 is \(2 \times 3 \times 5^2\).

Step-by-step solution

Find the prime factors

Start finding the smallest prime number which divides the integer 150 completely, that is remainder equals to zero. The smallest prime number is 2, but 2 does not divide 150 evenly, so try the next prime number, 3. It can divide 150 evenly, so we can say that 3 is a prime factor of 150. To find next prime factor, divide 150 by 3 to get the result 50.

Find the next prime factor

Again find the smallest prime number which divides the result, 50 completely. 2 can divide 50 evenly, so 2 is the another prime factor. If we divide 50 by 2, we get 25 as the result.

Find the remaining prime factors

Continue with this process and divide the resulting number by the smallest possible prime number. The smallest prime number by which 25 can be divided evenly is 5. 25 divided by 5 gives 5 which is also a prime number. So, 5 is the final prime factor we have, and 5 divided by 5 equals 1, at which point we can stop.

Key concepts

Prime Numbers

Prime numbers are the building blocks of whole numbers. These are numbers greater than 1 that have no divisors other than 1 and the number itself. For example, the prime factors of 150 include numbers like 2, 3, and 5.

Understanding prime numbers is crucial because they are used in various mathematical concepts, including prime factorization. They have unique properties, such as the fact that every number can be expressed as a product of prime numbers in a way that is exclusive to that number—a concept known as the Fundamental Theorem of Arithmetic.

Divisibility

Divisibility is about understanding which numbers can be divided by certain factors without leaving a remainder. For instance, in the factorization of the integer 150, we look for numbers that can divide 150 completely.

This process plays a vital role in finding prime factors and simplifying fractions. Knowing the basic divisibility rules—for example, a number is divisible by 2 if it is even, and by 3 if the sum of its digits is divisible by 3—helps in quickly identifying possible factors without performing long division.

Factorization Methods

There are several methods for factorizing integers, such as trial division, use of a factor tree, or application of more advanced methods like the sieve of Eratosthenes for larger numbers.

For example, in the factorization of 150, trial division is used, which is a systematic way of checking divisibility by prime numbers starting from the smallest. Understanding these methods is essential since factorization is a key step in simplifying algebraic expressions and solving equations.

Integer Factorization

Integer factorization is the process of breaking down a composite number into its prime factors, as seen in the sample exercise with the number 150. Through this process, we have found that 150 can be expressed as the product of prime numbers 3, 2, and 5, with 5 appearing twice (as 5 squared).

This process is particularly useful not just for simplifying mathematical expressions, but also in various applications such as cryptography, where the security of encryption methods often relies on the difficulty of factoring large integers.