**Number theory**, or the study of

**integers**(the counting number 1, 2, 3..., their opposites –1, –2, –3..., and also 0), has fascinated mathematicians for years.

**Prime numbers**, a concept introduced to many students in grades 4 and up, are an essential to number theory. They kind the basic building blocks for every integers.

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A element number is a counting number that only has actually two factors, itself and one. Counting number which have much more than two components (such together 6, whose factors are 1, 2, 3, and 6), are claimed to be **composite numbers**. The number 1 only has actually one factor and usually isn"t taken into consideration either element or composite.

*Key standard: identify whether a given number is prime or composite, and also find all determinants for a whole number. (Grade 4)*

## Why perform Prime components Matter?

It"s the age-old inquiry that math teachers everywhere must compete with. *When will certainly I use this? *One notable instance is through *cryptography*, or the examine of creating and deciphering codes. Through the assist of a computer, it is straightforward to multiply 2 prime numbers. However, it deserve to be *extremely* daunting to aspect a number. Because of this, when a website sends and also receives details securely—something especially important for financial or medical websites, because that example—you deserve to bet there room prime numbers behind the scenes. Element numbers additionally show increase in a range of how amazing contexts, including physics, music, and even in the come of cicadas!

There is an additional place where prime numbers present up often, and also it"s straightforward to skipping when discussing applications: *math!* The study of pure mathematics is a topic that civilization practice, study, and share without worrying around where rather it might apply, comparable to just how a musician go not have to ask just how music applies to the actual world. Number theory is an extremely rich topic that is main to college courses, research papers, and also other branches that mathematics. Mathematicians of all stripes no doubt encounter number theory many times follow me their academic and skilled journeys.

## Writing a Product of element Factors

When a composite number is created as a product of every one of its element factors, we have actually the **prime factorization** that the number. Because that example, we deserve to write the number 72 together a product of element factors: (72=2^3 cdot 3^2). The expression (2^3 cdot 3^2) is stated to it is in the prime factorization that 72. The** basic Theorem of Arithmetic** states that every composite number deserve to be factored uniquely (except for the order of the factors) right into a product of prime factors. What this means is that just how you choose to element a number right into prime determinants makes no difference. As soon as you are done, the prime factorizations are essentially the same.

Examine the two **factor trees** because that 72 presented below.

When we get done factoring utilizing either set of components to start with, we still have actually three components of 2 and also two components of 3, or (2^3 cdot 3^2). This would certainly be true if we had actually started to aspect 72 as 24 times 3, 4 time 18, or any kind of other pair of factors for 72.

Knowing rules for divisibility is advantageous when factoring a number. For example, if a entirety number ends in 0, 2, 4, 6, or 8, we could constantly start the factoring process by separating by 2. It must be noted that due to the fact that 2 only has actually two factors, 1 and also 2, that is the only also prime number.

Another means to aspect a number other than using factor trees is to start dividing by prime numbers:

Once again, we can see that (72=2^3 cdot 3^2).

Also crucial to writing the element factorization that a number is an understanding of **exponents**. An exponent speak how countless times the basic is supplied as a factor. In the prime factorization that (72=2^3 cdot 3^2), the 2 is offered as a variable three times and the 3 is provided as a aspect twice.

There is a strategy we can use to figure out even if it is a number is prime. Find the square source (with the aid of a calculator if needed), and only examine prime numbers much less than or equal to it. Because that example, to view if 131 is prime, since the square root is between 11 and 12, we only require to examine for divisibility by 2, 3, 5, 7, and 11. There is no need to inspect 13, because 132 = 169, which is higher than 131. This works because if a prime number better than 13 divided 131, then the other aspect would need to be much less than 13—which we"re currently checking!

## Introducing the Concept: Finding element Factors

Making certain your students" work-related is neat and also orderly will aid prevent them indigenous losing determinants when constructing factor trees. Have actually them examine their element factorizations by multiplying the components to see if they gain the initial number.

**Prerequisite an abilities and Concepts: **Students will need to know and be able to usage exponents. They also will discover it advantageous to recognize the rules of divisibility because that 2, 3, 4, 5, 9 and 10.

Write the number 48 on the board.

**Ask**:

*Who can provide me 2 numbers who product is 48?*Students should determine pairs the numbers prefer 6 and also 8, 4 and 12, or 3 and also 16. Take one of the bag of factors and create a variable tree for the prime factorization the 48 where all students deserve to see it.

**Ask**: *How countless factors that two space there?* *(4) **How carry out I refer that using an exponent?*Students need to say to write it together (2^4). If they don"t, repeat them that the *exponent* tells how plenty of times the *base* is taken as a factor. End up writing the element factorization on the board as (2^4 cdot 3). Next, discover the prime factorization because that 48 making use of a different collection of factors.**Ask: ***What do you notification about the prime factorization the 48 because that this collection of factors?*Students should notice that the prime factorization that 48 is (2^4 cdot 3) for both of them.**Say**: *There is a theorem in math that says as soon as we variable a number into a product of element numbers, it can only be done one way, no counting the stimulate of the factors.*Illustrate this principle by showing them that the prime factorization of 48 could additionally be composed as (3 cdot 2^4), but mathematically, that"s the same thing together (2^4 cdot 3).**Say**: *Now let"s try one on your own. Discover the prime factorization of 60 by developing a factor tree because that 60.*Have all students independently aspect 60. Together they complete their factorizations, observe what student do and take note of different approaches and visual representations. Ask for a college student volunteer to factor 60 because that the entire class to see.**Ask**: *Who factored 60 differently?*Have students that factored 60 in different ways (either by beginning with different determinants or by visually representing the element tree differently) display their job-related to the class. Ask college student to explain similarities and differences in the factorizations. If nobody used various factors, display the course a factorization that starts v a different collection of components for 60 and also have students identify similarities and also differences in between your factor tree and other students".**Ask**: *If I said the prime factorization of 36 is 22 • 9, would certainly I be right?*The students should say no, since 9 is no a element number. If they don"t, repeat them the the element factorization of a number way all the determinants must it is in prime and 9 is not a prime number.Place the complying with composite numbers on the board and ask lock to create the prime factorization because that each one using element trees: 24, 56, 63, and 46.

## Developing the Concept: Product of prime Numbers

Now that students can find the element factorization for numbers which are acquainted products, that is time because that them to usage their rules because that divisibility and other notions to uncover the element factorization of unfamiliar numbers. Create the number 91 on the board.

**Say**:

*Yesterday, we wrote some number in your prime administer form.*

**Ask**:

*Who have the right to write 91 together a product of element numbers?*Many students could say that can"t it is in done, since they will recognize that 2, 3, 4, 5, 9 and 10 don"t divide it. They may not shot to watch if 7 divides it, which it does. If they don"t identify that 7 divides 91, show it because that them. The element factorization that 91 is (7 cdot 13). Next, write the number 240 ~ above the board.

**Ask**:

*Who deserve to tell me two numbers who product is 240?*Students are most likely to say 10 and also 24. If not, ask lock to use their rules because that divisibility to view if castle can find two numbers. Produce a variable tree because that 240 prefer the one below.

**Ask**: *How countless factors of two room there in the element factorization that 240?* *(4) **Who have the right to tell me how to write the prime factorization the 240? (24 • 3 • 5)*Facilitate a discussion around different methods to factor 240 and the pros and cons of each method. If you begin with 2 and also 120, you finish up v the exact same prime administer in the end, but you finish up with a "one-sided tree" that part students may uncover more an overwhelming to job-related with. Have actually students identify ways the they like to factor and guide lock to describe their reasoning*.***Say**: *Since the prime factorization the 240 is 24 • 3 • 5, **the only prime number which divide this number space 2, 3 and also 5. Element numbers favor 7 and also 11 will not division the number, because they execute not appear in the prime factorization that the number.*Write the number 180 ~ above the board.**Ask**: *What 2 numbers might we start with to discover the prime factorization of 180?* *What other numbers could we use?*Encourage students to find a variety of pairs, such together 10 and 18 or 9 and 20. If nobody mentions either pair, indicate them both as possibilities. Have fifty percent the students use 10 and also 18 and also the other fifty percent use 9 and also 20. Have actually two students develop the two determinants for the class to see.* Ask: If the prime factorization the a number is* 22 • 5 • 7,

*what have the right to you phone call me around the number?*

**Ask**:

*If the prime factorization the a number is 3*

*3*

*• 11,*

*what can you tell me around this number?*Repeat the vault exercise v a brand-new number. Some feasible observations: because (3^2) is a factor, the number is divisible by 9 and also the sum of the number"s digits is a multiple of nine. Because the product of odd numbers is constantly odd, the number is one odd number. Castle might additionally tell you the it is a composite number, 5 is no a aspect of the number, and also so on.Give castle the complying with numbers and ask lock to uncover their prime factorization: 231, 117, and 175. Also give the following prime factorizations that numbers and also ask castle to write down at the very least two points they know about both the number represented: (3^2 cdot 5^2), (2^3 cdot 3 cdot 13), and (2^2 cdot 3 cdot 5). You can of course change both the numbers and also factorizations to complement what your students are all set for.

**Wrap-Up and Assessment Hints**

Finding the element factorization that numbers will certainly strengthen your students" simple facts and understanding of multiplication. College student who do not know their basic multiplication facts will likely struggle through this, because they do not recognize commodities such together 24 or 63 readily. Transforming the problem around and giving lock the element factorization the a number and also asking lock what castle know about the number there is no multiplying it out is a great way to assess their understanding of the divisibility rules, the ide of factoring, and multiplication in general.See more: What Does It Mean When You Dream About Your Old Crush Liking You

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*To construct students" conceptual understanding and assist them grow into procedurally fluently mathematicians, explore *HMH into Math*, our core equipment for** **K–8 mathematics instruction.*