**Number theory** has fascinated mathematicians for years. Fundamental to number theory are numbers themselves, and the basic building blocks for numbers are **prime numbers**. A prime number is a counting number that only has two factors, itself and one. Counting numbers which have more than two factors (such as six, whose factors are 1, 2, 3 and 6), are said to be **composite numbers**. The number one only has one factor and is considered to be neither prime nor composite.

When a composite number is written as a product of all of its prime factors, we have the prime factorization of the number. For example, the number 72 can be written as a product of primes as: 72 = 2^{3}**•**** **3^{2}. The expression "2^{3 }**•** 3^{2}" is said to be the **prime factorization** of 72. The Fundamental Theorem of Arithmetic states that every composite number can be factored uniquely (except for the order of the factors) into a product of prime factors. What this means is that how you choose to factor a number into prime factors makes no difference. When you are done, the prime factorizations are essentially the same. Examine the two **factor trees** for 72 given below.

When we get done factoring using either set of factors to start with, we still have three factors of two and two factors of three or 2^{3 }**•** 3^{2}. This would be true if we had started to factor 72 as 24 times 3, 4 times 18, or any other pair of factors for 72.

Knowing the rules for divisibility will be very helpful when seeking to write a number in prime factorization form. Since a number is divisible by two if it ends in either 0, 2, 4, 6, or 8, it should be noted that two is the only even prime number. Another way to factor a number other than using factor trees is to start dividing by prime numbers, as shown below.

Once again, we can see that 72 = 2^{3 }**•** 3^{2}. Another key idea in writing the prime factorization of a number is an understanding of **exponents**. An exponent tells how many times the base is used as a factor. In the prime factorization of 72 = 2^{3 }**•** 3^{2}, the two is used as a factor three times and the three is used as a factor twice.

When checking to see if a number is prime or not, you need only divide by those prime numbers which when squared remain less than the given number. For example to see if 131 is prime, you need only check for divisibility by 2, 3, 5, 7, and 11, since 13^{2} = 169. If a prime number greater than 13 divided 131, then the other factor would have to be less than 13 and you would have checked those already.^{}

## Introducing the Concept: Prime Factors

Making sure your students' work is neat and orderly will help prevent them from losing factors when constructing factor trees. Have them check their prime factorizations by multiplying the factors to see if they get the original number.

**Prerequisite Skills and Concepts: **Students will need to know and be able to use exponents. They also will find it helpful to know the rules of divisibility for 2, 3, 4, 5, 9 and 10.

Write the number 48 on the board.

**Ask**:*Who can give me two numbers whose product is 48?*

Students should identify pairs of numbers like 6 and 8, 4 and 12, or 3 and 16. Take one of the pairs of factors and create a factor tree for the prime factorization of 48 on the board or on an overhead transparency as shown below.

**Ask**:*How many factors of two are there?*(4)*How do I express that using an exponent?*

Students should say to write it as "2^{4}" If they don't, remind them that the exponent tells how many times the base is taken as a factor. Finish writing the prime factorization on the board as 2^{4}• 3.

Next, find the prime factorization for 48 using a different set of factors.

**Ask:**What do you notice about the prime factorization of 48 for this set of factors?

Students should notice that the prime factorization of 48 = 2^{4}x 3 for both of them.

**Say**:*There is a theorem in mathematics that says when we factor a number into a product of prime numbers, it can only be done one way, not counting the order of the factors*

Illustrate what is meant by that by showing them 12 = 2^{2 }• 3 OR 12 = 3 • 2^{2}.**Say**:*Now let's try one on your own. Find the prime factorization of 60 by creating a factor tree for 60.*Have someone come to the board and show how to find the prime factorization of 60.**Ask**:*Did anyone else start with a different set of factors for 60?*If they did, have them show their work as well. If not, show them by starting with a different set of factors for 60.**Ask**:*If I said the prime factorization of 36 is 2*The students should say no, because 9 is not a prime number. If they don't, remind them that the prime factorization of a number means all the factors must be prime and 9 is not a prime number.^{2}• 9, would I be right?

Place the following composite numbers on the board and ask them to write the prime factorization for each one using factor trees: 24, 56, 63, and 46.

**Say**:*Now that you have a good idea of what a number line with negative numbers looks like, each of you will make a number line using the strip of paper I will pass out to you.*

Tell them to write neatly so they can read their number lines and to be careful to place the numbers appropriately on their number line, so that it looks like the one up front.

## Developing the Concept: Prime Factors

Now that students can find the prime factorization for numbers which are familiar products, it is time for them to use their rules for divisibility and other notions to find the prime factorization of unfamiliar numbers. Write the number 91 on the board.

**Say**:*Yesterday, we wrote some numbers in their prime factorization form.***Ask**:*Who can write 91 as a product of prime numbers?*

Many students might say it can't be done, because they will recognize that 2, 3, 4, 5, 9 and 10 don't divide it. They may not try to see if 7 divides it, which it does. If they don't recognize that 7 divides 91, show them that 7 does divide it. The prime factorization of 91 is 7 • 13.Write the number 240 on the board.

**Ask**:*Who can tell me two numbers whose product is 240?*

Students will probably say 10 and 24. If not, ask them to use their rules for divisibility to see if they can find two numbers. Create a factor tree for 240 like the one below.

**Ask**:*How many factors of two are there in the prime factorization of 240?*(4)*Who can tell me how to write the prime factorization of 240? (2*^{4}• 3 • 5)^{}**Say**:*It was better to start this process with two factors like 10 and 24 or 20 and 12 than to take two factors like 2 and 120 or 3 and 80 because each of the previous numbers can be broken down and the end result will probably take fewer steps.***Say**:*Since the prime factorization of this number is**2*^{4}• 3 • 5,*the only prime numbers which divide this number are 2, 3 and 5. Prime numbers like 7 and 11 will not divide the number, because they do not appear in the prime factorization of the number.*Write the number 180 on the board.**Ask**:*What two numbers might we start with to find the prime factorization of 180?*(If they say 10 and 18, say that's good.)*What other numbers could we use?*If no one says 9 and 20, mention it as another possibility. Have half the students try use 10 and 18 and the other half use 9 and 20. Have two students put the two factor trees on the board for the class to see.**Ask**:*If the prime factorization of a number is*2^{2}• 5 • 7,*what can you tell me about the number?*They should say that the number is even and it ends in zero, since both 2 and 5 divide the number. They may also tell you other things, such as it is a composite number, it is greater than 100, that three is not a factor of the number, etc. Questions like the one above get at a depth of understanding about the prime factorization of a number.**Ask**:*If the prime factorization of a number is 3*^{3}*• 11,**what can you tell me about this number?*

The sum of its digits is a multiple of nine and the number is an odd number. They might also tell you that it is a composite number, five is not a factor of the number, etc.

Give them the following numbers and ask them to find their prime factorization: 231, 117, and 175. Also give the following prime factorizations of numbers and ask them to write down at least two things they know about both the number represented: 3^{2}• 5^{2}, 2^{3 }• 3 • 13, and 2^{2}• 3 • 5.^{}

**Wrap-Up and Assessment Hints**Finding the prime factorization of numbers will strengthen your students' basic facts and understanding of multiplication. Students who do not know their basic multiplication facts well may struggle with this, because they do not recognize products such as 24 or 63 readily. Turning the problem around and giving them the prime factorization of a number and asking them what they know about the number without multiplying it out is a good way to assess their understanding of the divisibility rules, the concept of factor, and multiplication in general.

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