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

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

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

Why carry out Prime factors Matter?

It"s the age-old concern that math teachers anywhere must compete with. When will certainly I usage this? One notable instance is with cryptography, or the examine of creating and also deciphering codes. V the assist of a computer, it is basic to multiply two prime numbers. However, it have the right to be extremely daunting to element a number. Because of this, as soon as a website sends and receives details securely—something specifically important for financial or clinical websites, for example—you can bet there are prime numbers behind the scenes. Prime numbers also show up in a selection of how amazing contexts, including physics, music, and even in the arrival of cicadas!

There is another place whereby prime numbers present up often, and also it"s simple to overlook when discussing applications: math! The examine of pure mathematics is a subject that civilization practice, study, and share without worrying around where rather it could apply, comparable to just how a musician does not need to ask exactly how music applies to the genuine world. Number theory is an extremely rich topic that is main to university courses, study papers, and also other branches the mathematics. Mathematicians of every stripes no doubt encounter number theory many times along their scholastic and skilled journeys.


Writing a Product of prime Factors

When a composite number is created as a product of every one of its element factors, we have actually the prime factorization of the number. Because that example, we deserve to write the number 72 as a product of element factors: \(72=2^3 \cdot 3^2\). The expression \(2^3 \cdot 3^2\) is said to be the prime factorization of 72. The fundamental Theorem of Arithmetic says that every composite number can be factored uniquely (except because that the bespeak of the factors) right into a product of prime factors. What this way is that just how you select to factor a number right into prime factors makes no difference. When you space done, the prime factorizations are essentially the same.

Examine the two factor trees for 72 shown below.

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

Knowing rules because that divisibility is useful when factoring a number. Because that example, if a whole number end in 0, 2, 4, 6, or 8, we could constantly start the factoring process by dividing by 2. It should be detailed that due to the fact that 2 only has two factors, 1 and also 2, that is the only also prime number.

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

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

Also an essential to writing the prime factorization that a number is an understanding of exponents. An exponent speak how numerous times the base is supplied as a factor. In the prime factorization that \(72=2^3 \cdot 3^2\), the 2 is provided as a element three times and the 3 is supplied as a element twice.

There is a strategy we have the right to use to number out whether a number is prime. Find the square source (with the help of a calculator if needed), and also only examine prime numbers much less than or same to it. Because that example, to watch if 131 is prime, because the square root is between 11 and 12, us only require to inspect for divisibility by 2, 3, 5, 7, and also 11. Over there is no require to examine 13, because 132 = 169, which is greater than 131. This works due to the fact that if a element number greater than 13 split 131, climate the other aspect would have to be much less than 13—which we"re already checking!

Introducing the Concept: Finding prime Factors

Making sure your students" job-related is neat and orderly will help prevent them indigenous losing factors when creating factor trees. Have actually them check their element factorizations by multiply the factors to watch if they gain the original number.

Prerequisite skills and Concepts: Students will must know and be may be to use exponents. They also will find it beneficial 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 give me two numbers who product is 48?Students should recognize pairs of numbers favor 6 and also 8, 4 and also 12, or 3 and 16. Take among the bag of factors and create a factor tree because that the element factorization of 48 wherein all students can see it.

Teaching Product Prime components Inline 3 element Tree
Ask: How plenty of factors that two space there? (4) How execute I express that using an exponent?Students need to say to create it together \(2^4\). If castle don"t, repeat them the the exponent tells how plenty of times the base is taken as a factor. Complete writing the prime factorization top top the board as \(2^4 \cdot 3\). Next, find the element factorization for 48 making use of a different collection of factors.Ask: What perform you an alert about the element factorization of 48 because that this set of factors?Students should notice that the element factorization of 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 right into a product of prime numbers, it have the right to only be done one way, not counting the bespeak of the factors.Illustrate this principle by mirroring them the the prime factorization that 48 could additionally be written as \(3 \cdot 2^4\), yet mathematically, that"s the very same thing together \(2^4 \cdot 3\).Say: Now let"s try one on your own. Find the prime factorization of 60 by producing a element tree because that 60.Have every students independently element 60. As they complete their factorizations, observe what student do and also take note of different approaches and also visual representations. Ask for a student volunteer to factor 60 because that the entire course to see.Ask: Who factored 60 differently?Have students that factored 60 in different way (either by starting with different determinants or by visually representing the element tree differently) show their occupational to the class. Ask college student to define similarities and differences in the factorizations. If nobody used various factors, show the course a factorization that starts v a different collection of factors for 60 and also have students recognize similarities and also differences between your aspect tree and other students".Ask: If I stated the element factorization of 36 is 22 • 9, would certainly I be right?The students have to say no, due to the fact that 9 is no a element number. If castle don"t, repeat them that the element factorization of a number method all the factors must it is in prime and also 9 is no a element number.Place the adhering to composite number on the board and ask them to create the prime factorization because that each one using variable trees: 24, 56, 63, and 46.

Developing the Concept: Product of element Numbers

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

Say: Yesterday, we created some numbers in their prime administer form.Ask: Who can write 91 together a product of element numbers?Many students might say that can"t it is in done, due to the fact that they will acknowledge that 2, 3, 4, 5, 9 and 10 don"t division it. They might not shot to check out if 7 divides it, which the does. If lock don"t recognize that 7 divides 91, show it for them. The element factorization of 91 is \(7 \cdot 13\). Next, create the number 240 top top the board.Ask: Who can tell me 2 numbers whose product is 240?Students are likely to say 10 and 24. If not, ask them to use their rules because that divisibility to view if lock can discover two numbers. Create a variable tree for 240 like the one below.

Teaching Product Prime determinants Inline 4 variable Tree
Ask: How many factors of two room there in the prime factorization of 240? (4) Who deserve to tell me just how to compose the prime factorization that 240? (24 • 3 • 5)Facilitate a discussion approximately different ways to variable 240 and the pros and cons of every method. If you begin with 2 and 120, you finish up through the exact same prime factorization in the end, but you end up with a "one-sided tree" that part students may discover more complicated to occupational with. Have students determine ways that they choose to factor and guide castle to describe their reasoning.Say: Since the element factorization of 240 is 24 • 3 • 5, the just prime numbers which division this number space 2, 3 and 5. Prime numbers prefer 7 and also 11 will certainly not division the number, due to the fact that they execute not show up in the prime factorization that the number.Write the number 180 top top the board.Ask: What two numbers can we start with to find the element factorization that 180? What various other numbers can we use?Encourage student to uncover a selection of pairs, such together 10 and 18 or 9 and also 20. If no one mentions one of two people pair, indicate them both together possibilities. Have half the students use 10 and also 18 and the other half use 9 and also 20. Have two students develop the two components for the class to see.Ask: If the prime factorization that a number is 22 • 5 • 7, what have the right to you phone call me about the number?Ask: If the element factorization of a number is 33 • 11, what can you tell me about this number?Repeat the vault exercise through a 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 that odd numbers is constantly odd, the number is one odd number. Lock might also tell you the it is a composite number, 5 is no a variable of the number, and also so on.Give castle the complying with numbers and also ask castle to find their prime factorization: 231, 117, and also 175. Also give the following prime factorizations the numbers and ask lock to write down at the very 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 adjust both the numbers and factorizations to match what your students are prepared for.

Wrap-Up and also Assessment Hints

Finding the prime factorization the numbers will certainly strengthen her students" basic facts and understanding the multiplication. Student who do not recognize their simple multiplication truth will most likely struggle through this, since they carry out not recognize commodities such as 24 or 63 readily. Transforming the problem around and also giving lock the prime factorization that a number and asking castle what castle know around the number without multiplying it the end is a good way to assess their understanding of the divisibility rules, the concept of factoring, and also multiplication in general.

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