Number theory, or the examine of integers (the counting number 1, 2, 3..., their opposites –1, –2, –3..., and 0), has fascinated mathematicians because that years. Prime numbers, a concept introduced to many students in grades 4 and also up, are fundamental to number theory. They kind the an easy building blocks for every integers.

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

Key standard: determine whether a given number is element or composite, and find all components for a totality number. (Grade 4)

Why carry out Prime factors Matter?

It"s the age-old concern that mathematics teachers anywhere must contend with. When will I use this? One notable example is v cryptography, or the study of creating and also deciphering codes. Through the help of a computer, that is basic to multiply 2 prime numbers. However, it have the right to be extremely challenging to element a number. Because of this, when a website sends and receives information securely—something especially important for financial or clinical websites, because that example—you have the right to bet there are prime numbers behind the scenes. Prime numbers additionally show increase in a variety of how amazing contexts, including physics, music, and also even in the come of cicadas!

There is another place where prime numbers present up often, and it"s straightforward to skip when discussing applications: math! The research of pure mathematics is a object that people practice, study, and also share without worrying around where rather it might apply, similar to exactly how a musician does not must ask how music uses to the actual world. Number concept is an extremely rich topic the is main to college courses, research papers, and other branches of mathematics. Mathematicians of all stripes no doubt conference number theory plenty of times follow me their academic and experienced journeys.

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Writing a Product of prime Factors

When a composite number is written as a product of every one of its prime factors, we have actually the prime factorization that the number. Because that example, we can write the number 72 together a product of prime factors: (72=2^3 cdot 3^2). The expression (2^3 cdot 3^2) is stated to it is in the prime factorization the 72. The an essential Theorem of Arithmetic states that every composite number have the right to be factored uniquely (except for the order of the factors) right into a product of prime factors. What this method is that just how you select to factor a number into prime factors makes no difference. When you space done, the element factorizations are basically the same.

Examine the 2 factor trees because that 72 presented below.


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

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

Another method to element a number various other than using element trees is to start separating by element numbers:


Teaching Product Prime determinants Inline 2 department Expressio
Once again, we deserve to see that (72=2^3 cdot 3^2).

Also key to creating the element factorization of a number is an knowledge of exponents. One exponent speak how many times the basic is provided as a factor. In the prime factorization that (72=2^3 cdot 3^2), the 2 is provided as a aspect three times and also the 3 is offered 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. Find the square source (with the help of a calculator if needed), and only examine prime numbers less than or equal to it. Because that example, to view if 131 is prime, due to the fact that the square source is in between 11 and also 12, we only need to check for divisibility by 2, 3, 5, 7, and also 11. There is no need to examine 13, since 132 = 169, i m sorry is higher than 131. This works since if a element number higher than 13 split 131, climate the other element would have to be much less than 13—which we"re already checking!


Introducing the Concept: Finding prime Factors

Making certain your students" work-related is neat and also orderly will aid prevent them native losing factors when creating factor trees. Have actually them inspect their element factorizations by multiply the factors to watch if they acquire the initial number.

Prerequisite an abilities and Concepts: Students will have to know and also be may be to usage exponents. They likewise will uncover it helpful to recognize the rules of divisibility for 2, 3, 4, 5, 9 and also 10.

Write the number 48 on the board.

Ask: Who can offer me two numbers who product is 48?Students should identify pairs of numbers choose 6 and also 8, 4 and 12, or 3 and also 16. Take among the pairs of factors and create a aspect tree because that the prime factorization the 48 whereby all students have the right to see it.

Teaching Product Prime components Inline 3 aspect Tree
Ask: How countless factors of two space there? (4) How perform I express that making use of an exponent?Students need to say to write it together (2^4). If they don"t, repeat them that the exponent speak how countless times the base is taken together a factor. Complete 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 set of factors.Ask: What execute you an alert about the element factorization that 48 for this collection of factors?Students should an alert that the prime factorization the 48 is (2^4 cdot 3) because that both the them.Say: There is a theorem in mathematics that says as soon as we aspect a number right into a product of prime numbers, it deserve to only be done one way, no counting the order of the factors.Illustrate this ide by mirroring them the the element factorization the 48 could likewise be composed as (3 cdot 2^4), but mathematically, that"s the very same thing together (2^4 cdot 3).Say: Now let"s try one on your own. Discover the element factorization the 60 by producing a variable tree for 60.Have every students independently element 60. Together they finish their factorizations, watch what students do and take note of various approaches and visual representations. Ask because that a student volunteer to factor 60 because that the entire course to see.Ask: Who factored 60 differently?Have students that factored 60 differently (either by beginning with different determinants or by visually representing the variable tree differently) show their job-related to the class. Ask student to describe similarities and also differences in the factorizations. If nobody used different factors, display the course a factorization the starts through a different set of factors for 60 and also have students determine similarities and differences between your variable tree and also other students".Ask: If I claimed the element factorization the 36 is 22 • 9, would certainly I be right?The students should say no, since 9 is no a prime number. If lock don"t, remind them the the prime factorization that a number way all the components must be prime and also 9 is no a element number.Place the complying with composite number on the board and also ask lock to create the element factorization for each one using element trees: 24, 56, 63, and also 46.


Developing the Concept: Product of prime Numbers

Now the students can discover the prime factorization for numbers i beg your pardon are familiar products, it is time because that them to use their rules for divisibility and also other notions to uncover the prime factorization that unfamiliar numbers. Compose the number 91 ~ above the board.

Say: Yesterday, we wrote some numbers in their prime factorization form.Ask: Who deserve to write 91 as a product of element numbers?Many students could say the can"t it is in done, because they will acknowledge that 2, 3, 4, 5, 9 and 10 don"t division it. They might not try to view if 7 divides it, which that 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, compose the number 240 ~ above the board.Ask: Who can tell me two numbers whose product is 240?Students are most likely to speak 10 and 24. If not, ask castle to use their rules because that divisibility to view if castle can discover two numbers. Develop a variable tree because that 240 favor the one below.

Teaching Product Prime determinants Inline 4 element Tree
Ask: How plenty of factors the two are there in the element factorization of 240? (4) Who can tell me exactly how to compose the element factorization that 240? (24 • 3 • 5)Facilitate a discussion about different means to aspect 240 and the pros and also cons of each method. If you start with 2 and also 120, you finish up through the same prime administer in the end, yet you end up with a "one-sided tree" that part students may uncover more an overwhelming to job-related with. Have actually students determine ways the they choose to factor and also guide lock to describe their reasoning.Say: Since the prime factorization that 240 is 24 • 3 • 5, the just prime numbers which divide this number room 2, 3 and 5. Element numbers favor 7 and also 11 will certainly not divide the number, because they carry out not show up in the element factorization the the number.Write the number 180 ~ above the board.Ask: What two numbers could we begin with to find the element factorization the 180? What other numbers could we use?Encourage college student to discover a range of pairs, such as 10 and also 18 or 9 and 20. If nobody mentions either pair, indicate them both together possibilities. Have half the students usage 10 and 18 and the other half use 9 and 20. Have two students develop the two factors for the course to see.Ask: If the prime factorization that a number is 22 • 5 • 7, what have the right to you tell me around the number?Ask: If the prime factorization that a number is 33 • 11, what can you phone call me around this number?Repeat the ahead exercise v a new number. Some feasible observations: due to the fact that (3^2) is a factor, the number is divisible by 9 and also the amount of the number"s number is a many of nine. Since the product of odd numbers is always odd, the number is an odd number. They might also tell you that it is a composite number, five is no a variable of the number, and so on.Give castle the complying with numbers and ask lock to find their element factorization: 231, 117, and 175. Also give the adhering to prime factorizations that numbers and also ask lock to create 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 adjust both the numbers and also factorizations to complement what her students are prepared for.

Wrap-Up and also Assessment Hints

Finding the element factorization of numbers will strengthen your students" basic facts and also understanding of multiplication. Student who do not know their an easy multiplication facts will most likely struggle with this, due to the fact that they do not recognize assets such as 24 or 63 readily. Turning the trouble around and also giving lock the element factorization the a number and asking them what they know about the number without multiplying it the end is a an excellent way to evaluate their expertise of the divisibility rules, the principle of factoring, and multiplication in general.

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