Number theory, or the research of integers (the counting numbers 1, 2, 3..., your opposites –1, –2, –3..., and also 0), has fascinated mathematicians for years. Prime numbers, a concept introduced to most students in qualities 4 and also up, are an essential to number theory. They type the basic building block 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 stated to it is in composite numbers. The number 1 only has actually one factor and usually isn"t thought about either element or composite.

Key standard: recognize whether a provided number is prime or composite, and find all components for a totality number. (Grade 4)

## Why perform Prime determinants Matter?

It"s the age-old inquiry that math teachers almost everywhere must compete with. When will certainly I use this? One notable example is v cryptography, or the examine of creating and also deciphering codes. With the help of a computer, it is basic to multiply two prime numbers. However, it deserve to be extremely complicated to aspect a number. Since of this, as soon as a website sends and receives info securely—something especially important because that financial or clinical websites, for example—you have the right to bet there room prime numbers behind the scenes. Prime numbers also show up in a variety of surprising contexts, consisting of physics, music, and even in the come of cicadas!

There is an additional place wherein prime numbers show up often, and it"s basic to overlook when discussing applications: math! The examine of pure math is a topic that people practice, study, and also share without worrying around where rather it could apply, comparable to just how a musician does not have to ask how music uses to the real world. Number theory is really rich topic the is central to university courses, research study papers, and also other branches that mathematics. Mathematicians of all stripes no doubt encounter number theory countless times follow me their academic and skilled journeys.

## Writing a Product of element Factors

When a composite number is composed as a product of all of its element factors, we have actually the prime factorization the the number. Because that example, we can write the number 72 as a product of prime factors: $$72=2^3 \cdot 3^2$$. The expression $$2^3 \cdot 3^2$$ is said to be the element factorization that 72. The basic Theorem of Arithmetic claims that every composite number have the right to be factored uniquely (except because that the order of the factors) into a product of prime factors. What this method is that exactly how you pick to variable a number right into prime factors makes no difference. When you room done, the element factorizations are essentially the same.

Examine the 2 factor trees for 72 shown below.

When we obtain done factoring using either set of factors to begin with, we still have actually three components of 2 and also two determinants of 3, or $$2^3 \cdot 3^2$$. This would certainly be true if we had started to variable 72 together 24 time 3, 4 time 18, or any type of other pair of components for 72.

Knowing rules because that divisibility is advantageous when factoring a number. For example, if a whole number end in 0, 2, 4, 6, or 8, we could constantly start the factoring procedure by dividing by 2. It have to be noted that since 2 only has actually two factors, 1 and 2, it is the only even prime number.

Another way to factor a number various other than using factor trees is to start dividing by element numbers:

Once again, we have the right to see that $$72=2^3 \cdot 3^2$$.

Also an essential to composing the prime factorization the a number is an expertise of exponents. An exponent tells how many times the base is provided as a factor. In the element factorization of $$72=2^3 \cdot 3^2$$, the 2 is offered as a variable three times and also the 3 is used as a variable twice.

There is a strategy we deserve to use to figure out whether a number is prime. Uncover the square source (with the assist of a calculator if needed), and only inspect prime numbers much less than or equal to it. Because that example, to see if 131 is prime, since the square root is in between 11 and also 12, us only need to check for divisibility by 2, 3, 5, 7, and 11. Over there is no need to inspect 13, due to the fact that 132 = 169, which is greater than 131. This works since if a element number greater than 13 separated 131, climate the other aspect would need to be much less than 13—which we"re already checking!

## Introducing the Concept: Finding prime Factors

Making certain your students" occupational is neat and orderly will assist prevent them from losing components when creating factor trees. Have actually them inspect their element factorizations by multiplying the determinants to check out if they obtain the original number.

Prerequisite skills and Concepts: Students will need to know and be maybe to usage exponents. They also will find it useful to understand the rule of divisibility because that 2, 3, 4, 5, 9 and 10.

Write the number 48 top top the board.

Ask: Who can offer me two numbers whose product is 48?Students should recognize pairs of numbers like 6 and also 8, 4 and 12, or 3 and 16. Take among the bag of factors and also create a aspect tree for the prime factorization that 48 where all students deserve to see it.

Ask: How many factors the two room there? (4) How execute I refer that making use of an exponent?Students must say to write it as $$2^4$$. If castle don"t, remind them the the exponent speak how numerous times the base is taken together a factor. Finish writing the element factorization top top the board as $$2^4 \cdot 3$$. Next, uncover the element factorization because that 48 utilizing a different collection of factors.Ask: What do you an alert about the element factorization the 48 for this collection of factors?Students should notice that the element factorization of 48 is $$2^4 \cdot 3$$ for both of them.Say: There is a to organize in math that says once we element a number right into a product of prime numbers, it have the right to only be done one way, no counting the bespeak of the factors.Illustrate this ide by reflecting them the the element factorization the 48 could also be created as $$3 \cdot 2^4$$, but mathematically, that"s the same thing together $$2^4 \cdot 3$$.Say: Now let"s try one on her own. Find the element factorization that 60 by creating a factor tree because that 60.Have all students independently aspect 60. As they finish their factorizations, observe what students do and take keep in mind of different approaches and also visual representations. Ask for a college student volunteer to aspect 60 for the entire class to see.Ask: Who factored 60 differently?Have students who factored 60 in different way (either by beginning with different factors or through visually representing the variable tree differently) present their work to the class. Ask student to explain similarities and differences in the factorizations. If no one used different factors, present the course a factorization that starts with a different collection of determinants for 60 and have students identify similarities and differences in between your element tree and other students".Ask: If I claimed the element factorization that 36 is 22 • 9, would certainly I it is in right?The students must say no, due to the fact that 9 is no a prime number. If castle don"t, remind them the the element factorization that a number way all the factors must be prime and 9 is no a element number.Place the following composite numbers on the board and also ask lock to create the element factorization because that each one using factor trees: 24, 56, 63, and also 46.

## Developing the Concept: Product of prime Numbers

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

Say: Yesterday, we created some number in your prime administrate form.Ask: Who have the right to write 91 as a product of element numbers?Many students might say it can"t be done, due to the fact that they will acknowledge that 2, 3, 4, 5, 9 and 10 don"t divide it. They might not try to check out if 7 divides it, which it does. If castle don"t recognize that 7 divides 91, demonstrate it for them. The element factorization that 91 is $$7 \cdot 13$$. Next, compose the number 240 on the board.Ask: Who can tell me 2 numbers whose product is 240?Students are likely to speak 10 and 24. If not, ask them to usage their rules because that divisibility to see if castle can discover two numbers. Create a factor tree because that 240 favor the one below.

Ask: How countless factors the two room there in the prime factorization the 240? (4) Who deserve to tell me just how to compose the element factorization that 240? (24 • 3 • 5)Facilitate a discussion roughly different methods to element 240 and the pros and cons of every method. If you begin with 2 and also 120, you end up v the exact same prime factorization in the end, however you end up v a "one-sided tree" that some students may uncover more challenging to work-related with. Have students identify ways that they like to factor and guide them to define their reasoning.Say: Since the element factorization the 240 is 24 • 3 • 5, the just prime numbers which division this number room 2, 3 and also 5. Element numbers prefer 7 and 11 will not division the number, because they execute not appear in the element factorization that the number.Write the number 180 on the board.Ask: What two numbers can we begin with to uncover the element factorization the 180? What other numbers can we use?Encourage college student to uncover a range of pairs, such together 10 and 18 or 9 and 20. If nobody mentions one of two people pair, suggest them both together possibilities. Have half the students use 10 and also 18 and also the other fifty percent use 9 and 20. Have actually two students produce the two determinants for the class to see.Ask: If the prime factorization of a number is 22 • 5 • 7, what can you call me around the number?Ask: If the element factorization the a number is 33 • 11, what deserve to you tell me around this number?Repeat the previous exercise through a brand-new number. Some feasible observations: because $$3^2$$ is a factor, the number is divisible by 9 and the amount of the number"s digits is a multiple of nine. Since the product of odd number is constantly odd, the number is an odd number. They might likewise tell you the it is a composite number, 5 is no a aspect of the number, and also so on.Give castle the adhering to numbers and ask them to uncover their element factorization: 231, 117, and 175. Also give the following prime factorizations of numbers and ask lock to write down at least two points they know around both the number represented: $$3^2 \cdot 5^2$$, $$2^3 \cdot 3 \cdot 13$$, and also $$2^2 \cdot 3 \cdot 5$$. You have the right to of course change both the numbers and factorizations to match what her students are ready for.

## Wrap-Up and Assessment Hints

Finding the prime factorization of numbers will strengthen your students" simple facts and understanding the multiplication. Students who execute not understand their an easy multiplication truth will most likely struggle through this, due to the fact that they carry out not recognize products such together 24 or 63 readily. Turning the problem around and giving them the prime factorization that a number and also asking castle what castle know about the number without multiplying it out is a good way to assess their expertise of the divisibility rules, the ide of factoring, and multiplication in general.

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