Number theory, or the study of integers (the counting number 1, 2, 3..., their opposites –1, –2, –3..., and 0), has fascinated mathematicians for years. Prime numbers, a concept introduced to most students in grades 4 and up, are fundamental to number theory. They type the straightforward building blocks for all integers.

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A element number is a count number the only has two factors, itself and one. Counting number which have an ext than two factors (such together 6, whose factors are 1, 2, 3, and 6), are said to it is in composite numbers. The number 1 only has actually one factor and usually isn"t considered either element or composite.

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

Why perform Prime factors Matter?

It"s the age-old question that mathematics teachers everywhere must contend with. When will I usage this? One notable instance is through cryptography, or the examine of creating and deciphering codes. Through the assist of a computer, the is straightforward to multiply 2 prime numbers. However, it can be extremely an overwhelming to factor a number. Because of this, as soon as a website sends and also receives info securely—something especially important for financial or medical websites, for example—you have the right to bet there space prime number behind the scenes. Element numbers also show up in a selection of how amazing contexts, consisting of physics, music, and also even in the come of cicadas!

There is another place where prime numbers show up often, and also it"s easy to skip when discussing applications: math! The examine of pure mathematics is a subject that human being practice, study, and also share there is no worrying around where else it could apply, similar to just how a musician walk not have to ask just how music applies to the real world. Number theory is an extremely rich topic the is main to university courses, study papers, and also other branches of mathematics. Mathematicians of all stripes no doubt encounter number theory numerous times along their academic and expert journeys.


Writing a Product of prime Factors

When a composite number is composed as a product of every one of its prime 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 said to be the element factorization the 72. The basic Theorem of Arithmetic states 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 means is that exactly how you select to variable a number into prime determinants makes no difference. Once you space done, the prime factorizations are essentially the same.

Examine the two factor trees because that 72 presented below.

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

Knowing rules because that divisibility is beneficial when factoring a number. Because that example, if a entirety number ends in 0, 2, 4, 6, or 8, us could always start the factoring procedure by separating by 2. It have to be provided that due to the fact that 2 only has two factors, 1 and also 2, the is the only also prime number.

Another way to variable a number various other than using variable trees is come start splitting by prime numbers:

Teaching Product Prime determinants Inline 2 division Expressio
Once again, we have the right to see that (72=2^3 cdot 3^2).

Also key to creating the element factorization the a number is an knowledge of exponents. One exponent speak how numerous times the basic is supplied as a factor. In the element factorization that (72=2^3 cdot 3^2), the 2 is used as a variable three times and the 3 is supplied as a aspect twice.

There is a strategy we have the right to use to number out even if it is a number is prime. Uncover the square root (with the assist of a calculator if needed), and only check prime numbers less than or same to it. For example, to check out if 131 is prime, due to the fact that the square source is in between 11 and also 12, we only need to inspect for divisibility by 2, 3, 5, 7, and also 11. There is no require to inspect 13, since 132 = 169, i m sorry is higher than 131. This works due to the fact that if a prime number better than 13 divided 131, then the other variable would have to be less than 13—which we"re already checking!

Introducing the Concept: Finding prime Factors

Making sure your students" work is neat and also orderly will assist prevent them indigenous losing components when constructing factor trees. Have them inspect their prime factorizations by multiplying the components to check out if they gain the initial number.

Prerequisite an abilities and Concepts: Students will should know and be maybe to usage exponents. They likewise will find it beneficial to know the rules of divisibility for 2, 3, 4, 5, 9 and also 10.

Write the number 48 ~ above the board.

Ask: Who can give me two numbers whose product is 48?Students should identify pairs that numbers like 6 and 8, 4 and also 12, or 3 and 16. Take one of the pairs of factors and create a factor tree for the element factorization that 48 wherein all students deserve to see it.

Teaching Product Prime components Inline 3 element Tree
Ask: How plenty of factors of two are there? (4) How perform I to express that using an exponent?Students must say to create it together (2^4). If they don"t, remind them the the exponent tells how many times the base is taken as a factor. Complete writing the element factorization ~ above the board together (2^4 cdot 3). Next, uncover the prime factorization for 48 making use of a different set of factors.Ask: What perform you an alert about the prime factorization of 48 because that this collection of factors?Students should notice that the element factorization that 48 is (2^4 cdot 3) for both that them.Say: There is a theorem in math that says when we variable a number right into a product of element numbers, it can only be done one way, no counting the bespeak of the factors.Illustrate this concept by showing them the the prime factorization that 48 could also be composed as (3 cdot 2^4), however mathematically, that"s the very same thing as (2^4 cdot 3).Say: Now let"s shot one on your own. Uncover the prime factorization that 60 by developing a element tree for 60.Have all students independently element 60. As they finish their factorizations, observe what college student do and also 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 components or through visually representing the aspect tree differently) present their job-related to the class. Ask college student to describe similarities and also differences in the factorizations. If nobody used different factors, show the class a factorization that starts through a different collection of factors for 60 and also have students identify similarities and differences in between your element tree and also other students".Ask: If I stated the prime factorization the 36 is 22 • 9, would certainly I be right?The students have to say no, since 9 is not a element number. If lock don"t, repeat them the the prime factorization the a number method all the determinants must it is in prime and 9 is no a element number.Place the adhering to composite numbers on the board and also ask them to compose the element factorization for each one using element trees: 24, 56, 63, and also 46.

Developing the Concept: Product of element Numbers

Now that students can discover the element factorization because that numbers i m sorry are familiar products, the is time for them to use their rules because that divisibility and also other notions to uncover the element factorization that unfamiliar numbers. Write the number 91 on the board.

Say: Yesterday, we wrote some numbers in your prime administrate form.Ask: Who have the right to write 91 together a product of element numbers?Many students might say that can"t be done, due to the fact that they will acknowledge that 2, 3, 4, 5, 9 and 10 don"t division it. They may not shot to watch if 7 divides it, which that does. If lock don"t identify that 7 divides 91, show it for them. The element factorization of 91 is (7 cdot 13). Next, compose the number 240 top top the board.Ask: Who have the right to tell me two numbers who product is 240?Students are most likely to to speak 10 and 24. If not, ask them to usage their rules because that divisibility to check out if lock can discover two numbers. Produce a element tree for 240 prefer the one below.

Teaching Product Prime determinants Inline 4 variable Tree
Ask: How many factors that two space there in the prime factorization of 240? (4) Who deserve to tell me how to create the element factorization that 240? (24 • 3 • 5)Facilitate a discussion approximately different methods to variable 240 and the pros and also cons of each method. If you start with 2 and 120, you end up v the exact same prime administer in the end, but you end up with a "one-sided tree" that some students may discover more complicated to work with. Have students determine ways that they like to factor and guide castle to explain their reasoning.Say: Since the prime factorization the 240 is 24 • 3 • 5, the just prime numbers which divide this number space 2, 3 and also 5. Element numbers prefer 7 and 11 will certainly not division the number, because they do not appear in the prime factorization the the number.Write the number 180 top top the board.Ask: What two numbers could we begin with to discover the prime factorization the 180? What various other numbers can we use?Encourage students to find a selection of pairs, such as 10 and also 18 or 9 and also 20. If nobody mentions either pair, imply them both as possibilities. Have fifty percent the students use 10 and also 18 and also the other half use 9 and also 20. Have actually two students create the two determinants for the class to see.Ask: If the prime factorization that a number is 22 • 5 • 7, what deserve to you tell me around the number?Ask: If the element factorization of a number is 33 • 11, what deserve to you phone call me around this number?Repeat the previous exercise v a brand-new number. Some feasible observations: since (3^2) is a factor, the number is divisible through 9 and also the sum of the number"s digits is a lot of of nine. Since the product of odd numbers is constantly odd, the number is one odd number. Lock might additionally tell you the it is a composite number, 5 is not a variable of the number, and also so on.Give them the complying with numbers and also ask lock to uncover their prime factorization: 231, 117, and also 175. Also give the adhering to prime factorizations of numbers and also ask lock to write down at the very least two things 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 your students are prepared for.

Wrap-Up and Assessment Hints

Finding the prime factorization the numbers will certainly strengthen her students" straightforward facts and understanding that multiplication. Student who do not understand their simple multiplication truth will likely struggle v this, because they perform not recognize products such as 24 or 63 readily. Turning the trouble around and giving them the element factorization the a number and also asking lock what lock know around the number there is no multiplying it the end is a good way to assess their knowledge of the divisibility rules, the principle of factoring, and also multiplication in general.

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