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.

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:

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.

## 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.