Number theory, or the examine of integers (the counting numbers 1, 2, 3..., their opposites –1, –2, –3..., and 0), has actually fascinated mathematicians because that years. Prime numbers, a ide introduced to most students in grades 4 and also up, are an essential to number theory. They type the straightforward building block for all integers.

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

Key standard: recognize whether a offered 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 concern that mathematics teachers almost everywhere must contend with. When will certainly I usage this? One notable instance is v cryptography, or the study of creating and deciphering codes. Through the assist of a computer, it is straightforward to multiply 2 prime numbers. However, it have the right to be extremely difficult to aspect a number. Due to the fact that of this, as soon as a website sends and receives info securely—something specifically important because that financial or clinical websites, for example—you can bet there space prime number behind the scenes. Prime numbers likewise show increase in a selection of surprising contexts, consisting of physics, music, and even in the come of cicadas!

There is another place wherein prime numbers show up often, and it"s basic to skipping when pointing out applications: math! The study of pure math is a object that world practice, study, and share without worrying around where rather it might apply, comparable to exactly how a musician walk not have to ask just how music applies to the real world. Number concept is an extremely rich topic that is main to university courses, research study papers, and also other branches that mathematics. Mathematicians of all stripes no doubt conference number theory numerous times along their scholastic and expert journeys.


Writing a Product of element Factors

When a composite number is created as a product of all of its element factors, we have actually the prime factorization that the number. For 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 claimed to be the element factorization the 72. The basic Theorem the Arithmetic states that every composite number can be factored unique (except for the order of the factors) into a product of prime factors. What this way is that how you select to aspect a number into prime determinants makes no difference. As soon as you room done, the element factorizations are basically the same.

Examine the 2 factor trees for 72 presented below.

New variable Tree
When we gain done factoring utilizing either set of determinants to begin with, us still have actually three components of 2 and two determinants of 3, or \(2^3 \cdot 3^2\). This would be true if we had actually started to aspect 72 together 24 time 3, 4 times 18, or any kind of other pair of determinants for 72.

Knowing rules for divisibility is advantageous when factoring a number. Because that example, if a whole number ends in 0, 2, 4, 6, or 8, us could always start the factoring process by separating by 2. It have to be detailed that since 2 only has actually two factors, 1 and also 2, that is the only even prime number.

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

Teaching Product Prime components Inline 2 division Expressio
Once again, we deserve to see the \(72=2^3 \cdot 3^2\).

Also key to composing the element factorization of a number is an understanding of exponents. One exponent tells how numerous times the basic is used as a factor. In the element factorization the \(72=2^3 \cdot 3^2\), the 2 is provided as a variable three times and the 3 is supplied as a variable twice.

There is a strategy we deserve to use to figure out even if it is a number is prime. Discover the square root (with the help of a calculator if needed), and also only inspect prime numbers less than or same to it. Because that example, to watch if 131 is prime, since the square source is between 11 and also 12, us only need to examine for divisibility by 2, 3, 5, 7, and 11. Over there is no require to examine 13, because 132 = 169, which is better than 131. This works because if a prime number greater than 13 separated 131, then the other variable would need to be less than 13—which we"re currently checking!

Introducing the Concept: Finding element Factors

Making sure your students" job-related is neat and orderly will aid prevent them indigenous losing components when creating factor trees. Have actually them check their prime factorizations by multiply the components to view if they get the initial number.

Prerequisite an abilities and Concepts: Students will should know and also be may be to usage exponents. They likewise will uncover it valuable to understand the rule of divisibility for 2, 3, 4, 5, 9 and also 10.

Write the number 48 ~ above the board.

Ask: Who can offer me 2 numbers who product is 48?Students should recognize pairs that numbers like 6 and 8, 4 and also 12, or 3 and 16. Take among the bag of factors and also create a element tree because that the prime factorization of 48 wherein all students deserve to see it.

Teaching Product Prime components Inline 3 factor Tree
Ask: How many factors that two room there? (4) How do I express that using an exponent?Students have to say to write it together \(2^4\). If lock don"t, repeat them the the exponent speak how plenty of times the base is taken together a factor. Finish writing the element factorization ~ above the board together \(2^4 \cdot 3\). Next, uncover the element factorization because that 48 making use of a different set of factors.Ask: What carry out you an alert about the prime factorization that 48 because that this set of factors?Students should notification that the element factorization the 48 is \(2^4 \cdot 3\) because that both of them.Say: There is a organize in math that says once we variable a number into a product of element numbers, it deserve to only be excellent one way, not counting the stimulate of the factors.Illustrate this concept by mirroring them the the prime factorization the 48 could additionally be composed as \(3 \cdot 2^4\), yet mathematically, that"s the same thing as \(2^4 \cdot 3\).Say: Now let"s shot one on her own. Discover the prime factorization the 60 by producing a aspect tree for 60.Have all students independently factor 60. Together they complete their factorizations, observe what college student do and also take note of various approaches and visual representations. Ask for a college student volunteer to aspect 60 for the entire course to see.Ask: Who factored 60 differently?Have students who factored 60 in different way (either by beginning with different components or by visually representing the factor tree differently) show their work-related to the class. Ask college student to describe similarities and differences in the factorizations. If nobody used various factors, display the course a factorization the starts with a different set of components for 60 and also have students identify similarities and also differences in between your element tree and also other students".Ask: If I claimed the prime factorization the 36 is 22 • 9, would certainly I it is in right?The students have to say no, because 9 is no a element number. If lock don"t, repeat them the the element factorization of a number method all the factors must it is in prime and 9 is no a prime number.Place the following composite numbers on the board and ask them to compose the element factorization for each one using aspect trees: 24, 56, 63, and 46.

Developing the Concept: Product of prime Numbers

Now that students can find the element factorization for numbers i beg your pardon are acquainted products, it is time because that them to use their rules because that divisibility and also other notions to find the element factorization of unfamiliar numbers. Write the number 91 ~ above the board.

Say: Yesterday, we created some number in your prime factorization form.Ask: Who deserve to write 91 together a product of element numbers?Many students can say that can"t be done, since they will recognize that 2, 3, 4, 5, 9 and also 10 don"t division it. They may not shot to see if 7 divides it, which the does. If lock don"t acknowledge that 7 divides 91, show it for them. The prime factorization the 91 is \(7 \cdot 13\). Next, write the number 240 top top the board.Ask: Who have the right to 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 for divisibility to see if they can uncover two numbers. Produce a factor tree because that 240 choose the one below.

Teaching Product Prime determinants Inline 4 factor Tree
Ask: How countless factors of two are there in the element factorization the 240? (4) Who deserve to tell me just how to create the prime factorization of 240? (24 • 3 • 5)Facilitate a discussion around different ways to aspect 240 and the pros and cons of every method. If you start with 2 and 120, you end up with the exact same prime administrate in the end, but you finish up v a "one-sided tree" that some students may find more an overwhelming to job-related with. Have students identify ways the they prefer to factor and also guide them to define their reasoning.Say: Since the element factorization the 240 is 24 • 3 • 5, the only prime numbers which divide this number are 2, 3 and also 5. Prime numbers prefer 7 and also 11 will not division the number, since they execute not appear in the prime factorization the the number.Write the number 180 top top the board.Ask: What two numbers might we start with to discover the element factorization that 180? What various other numbers might we use?Encourage students to uncover a range of pairs, such together 10 and also 18 or 9 and also 20. If no one mentions one of two people pair, suggest them both as possibilities. Have half the students use 10 and also 18 and the other half use 9 and also 20. Have two students create the two factors for the class to see.Ask: If the prime factorization of a number is 22 • 5 • 7, what can you tell me about the number?Ask: If the element factorization of a number is 33 • 11, what have the right to you tell me about this number?Repeat the previous exercise with a brand-new number. Some feasible observations: because \(3^2\) is a factor, the number is divisible by 9 and also the amount of the number"s number is a lot of of nine. Because the product the odd number is constantly odd, the number is one odd number. Castle might likewise 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 prime factorization: 231, 117, and 175. Additionally give the adhering to prime factorizations of numbers and ask them to write down at least two things they know about both the number represented: \(3^2 \cdot 5^2\), \(2^3 \cdot 3 \cdot 13\), and also \(2^2 \cdot 3 \cdot 5\). You can of course readjust both the numbers and factorizations to enhance what your students are ready for.

Wrap-Up and Assessment Hints

Finding the element factorization of numbers will certainly strengthen her students" simple facts and understanding of multiplication. Student who do not understand their basic multiplication facts will likely struggle with this, because they execute not recognize products such together 24 or 63 readily. Turning the trouble around and giving them the prime factorization of a number and also asking castle what they know about the number there is no multiplying it out is a great way to assess their expertise of the divisibility rules, the ide of factoring, and multiplication in general.

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