Number theory—the research of creature numbers—has fascinated mathematicians for years. Basic to number concept are entirety numbers: 0, 1, 2, 3, and also so on. The straightforward building blocks because that all entirety numbers room prime numbers. A prime number is a whole number that has only 2 factors: itself and one. As soon as a number has more than two factors, that is said to it is in a composite number.

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The numbers one and also zero space unusual cases. The number one has only one factor and is taken into consideration to be neither prime nor composite. The number zero has actually an infinite number of factors, together zero can be divided evenly by anything except zero. Because that this reason, zero is likewise considered no prime nor composite.

Check your Understanding:

Is 36 a element number or composite number? (Composite, because it has numerous factors various other than 1 and 36: 2, 3, 4, 6, 9, 12, and 18.)What are every one of the element numbers less than 10? (2, 3, 5, and 7.)
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Prime factorization of 36

Students in qualities 4 and up are frequently ready come learn about prime factors, and also students in qualities 6 and also up are generally ready to inspection the prime factorization that a number and explore exactly how it deserve to be offered to generate all entirety numbers.

Key Standards:

Find aspect pairs and also recognize the a whole number is a many of every of that is factors.Write and evaluate number expressions that stand for prime factorizations and also involve whole-number exponents.

When a composite number is created as a product of all of its element factors, we have actually the prime factorization the the number. Thus, if we create 36 together a product of all of its prime factors, us can uncover the prime factorization the 36.

We deserve to write 36 together a product of prime factors: 36 = 2² × 3². The expression 2² × 3² is claimed to it is in the prime factorization of 36.

The fundamental Theorem of Arithmetic

The Fundamental organize of Arithmetic states that every entirety number can be factored uniquely (except for the bespeak of the factors) right into a product of element factors. For example, there are numerous whole numbers that have the right to divide 36: 2, 3, 4, 6, 9, 12, and also 18. But no issue which element you begin with, once the number 36 is factored completely, the will always look like 2² × 3². The order in i beg your pardon you variable 36 doesn"t matter. Examine the two factor trees of 36 offered below.


Rime factor Tree administrate 36 picture 1
In the end, it didn"t matter if we began factoring 36 making use of 2 × 18 or 4 × 9. In the end, us still have two components of 2 and also two determinants of 3, or altogether, 2² × 3². This would be true if we had started to factor 36 as 12 time 3 or 6 time 6 instead.

Knowing the rules for divisibility will be very helpful when seeking to create a number in element factorization form. A dominance for checking if a number is divisible by 2 is by seeing if it ends in 0, 2, 4, 6, or 8. An alert that two is the only also prime number.

Dividing by element Numbers

Another means to factor a number various other than using aspect trees is to start dividing by element numbers, as displayed below.


Rime factor Tree factorization 36 image 2
Using this method, the element factorization is discovered by spring at every one of the numbers external the division symbols. As soon as again, we can see that 36 = 2² × 3². An alert the crucial importance of exponents in composing the element factorization the a number. One exponent speak how countless times the basic is used as a factor. In the prime factorization of 36 = 2² × 3², both the the determinants 2 and also 3 have an exponent that two because each factor shows up twice.

It is notoriously difficult to recognize whether a number is prime, particularly when the number is an extremely large. One helpful strategy to check whether a number is element or no is come only shot dividing by element numbers that, when squared, remain less than the original number. For example, to watch if 131 is prime, you need only check for divisibility through 2, 3, 5, 7, and also 11, since 13² = 169, and 169 is higher than 131. This strategy works due to the fact that if 131 to be divisible by a element number greater than 13, climate the other factor would need to be less than 13 and you would certainly have currently checked it.

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