Prime Factorization

Prime factorization. In our interconnected digital world, where every keystroke echoes through the vast expanse of cyberspace, we walk a challenging tightrope, exposing ourselves to the ever-lurking shadows of hackers, ready to exploit our vulnerabilities and breach the walls of our digital existence. Cryptography protects our data by encrypting codes. Prime factorization is a mathematical concept that helps with encryption, making it difficult for others to break those secret codes. Remember, we classify natural numbers as either composite or prime. Prime factorization is finding which prime numbers multiply together to make the original number. We can organize this process by creating a factor tree. A factor tree is a visual representation that helps us find the prime factors of a number. It is a way to break down a number by repeatedly dividing it into smaller factors. Let's use the number twenty-four as an example. To find the prime factorization of twenty-four, we look for the prime numbers that can be divided evenly into it. We can start by dividing it by the smallest prime number, which is two. Two times twelve makes twenty-four, so we write that here. Since two is prime, we can circle the number to stop here. Let's continue to divide twelve with two and six. Circle this two, and continue to divide six. The two and three will go here, and since they are prime, we will circle them. There are no more composite numbers, so we will list all the prime numbers in sequential order. Twenty-four is equal to two times two, times two, times three. Let's practice with the number, seventy-two. What are two factors of seventy -two? Eight and nine. Let's start with eight. What two factors make eight? Four and two. Two is prime, so we'll circle it and stop. Four is further broken down into two and two. Now, move over to nine. We have three and three, so we can circle and stop. List all the prime numbers in order. Seventy-two is equal to two times two times two times three times three. As you can see, prime factorization can get really long. We can write these expressions in a shorthand way, using exponents. Two is multiplied three times, so we can rewrite it as two to the third power, and the three is multiplied twice, so we can put three to the power of two. Here's one to try on your own. Find the prime factorization of ninety, and write the final expression using exponents if needed. Pause the video for extended time and resume when you're ready to review. There are several factors that you could choose to start with, but we will begin with nine times ten. Nine is made by three and three, which are both prime. And ten is made by two and five, also both prime, so we can stop here. Our prime numbers are two, three, three, and five, which we can write as two times three to the second power times five. Prime factorization is finding which prime numbers multiply together to make the original number. We can use prime factorization to simplify fractions, find the greatest common factor or least common multiple of numbers, test larger numbers for divisibility, and protect our data across the internet.