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Prime Factorization No, not this kind of prime.

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What is a prime? A prime is a natural number greater than 1 that has no positive divisors other than 1 and itself. In other words, see what you can divide the number by. If the only answers are the number itself and one, then the number is prime.

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And if it’s not prime? If a number is not prime it is called composite. 6 is divisible by 1, 2, 3, and 6. This means that 6 is a composite number.

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**So what is prime factorization?**

This is a way to break a number down into parts where each part is a prime number.

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**Let’s look at the number 6.**

6 can be broken down into 3 x 2 6 3 x 2

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**For 6 we write the prime factorization as**

6 = 3 x 2

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**How do we find the prime factorization?**

Division by primes Factor Tree

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Division by Primes The first method is called division by primes (or the birthday cake method) Let’s use the number 24 First we pick a prime that we can divide 24 by. Since it is even we will try 2.

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**2 3 6 2 12 2 24 First we divide 24 by 2 and the result is 12**

Next we choose another prime to now divide 12 by We divide 12 by 2 and the result is 6 Now we choose a prime to divide 6 by, let’s choose 3 this time The result is 2, which is a prime, once a prime answer is reached we stop Now to get our answer we look on the outside of the long division 2 3 6 2 12 2 24 The prime factorization of 24 is 2 x 2 x 2 x 3

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**The advantages to this method are in the ease of choosing your**

numbers. You must choose a prime number to divide by each time, which limits your choices to a few small numbers usually. The reason this is sometimes called the birthday cake method is because of the resulting shape. 2 3 6 2 12 2 24

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**Factor Tree The second method is called a factor tree**

Again we will use the number 24 This time we are not limited to our choice of numbers to divide by so we will pick 8

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**24 8 3 4 2 2 2 First we divide 24 by 8. The result is 3.**

i.e. 8 x 3 = 24 24 Next we look at 8. We can divide 8 by 4 to get a result of 2. 8 3 Then we look at the 3. This is a prime so we do not touch the 3. We will circle it to indicate we are done with it. The 4 can turn into 2 x 2. 4 2 The 2 is prime so we circle it. The last two 2’s are prime, so again we circle them. 2 2 Now look at all of our circled numbers. This is our prime factorization. 24 = 3 x 2 x 2 x 2

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The advantage to the factor tree method is the ability to choose from any number. (as long you can divide the original number evenly) 24 8 3 This method is named for the resemblance the final results have to a pine tree. 4 2 2 2

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**Which method should we use?**

Both division by primes and factor tree may be used in any case; however, one method may make the problem much easier. Often it is best to use division by primes when it is difficult to see large numbers that can be used. Factor tree is a good choice when it is easy to see a large number to divide by. For example, numbers ending in zero (divide by ten).

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Decomposing a Number into Prime Factors. Learning Goals We will use our divisibility rules so that we can decompose numbers into prime factors. We’ll.

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