In this explainer, we will certainly learn how to use different strategies to find prime factorization making use of exponents.

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You should currently understand the state prime number, composite number, and factor.

### Definition: Prime, Composite, and also Factor

Factors of a number room the numbers us multiply together to make the initial number. We regularly write determinants in pairs.**because that example, 6×1=6 and also 2×3=6, for this reason 1, 2, 3, and 6 are determinants of 6. One element pair for 6 is 1 and 6 since their product is 6; another factor pair is 2 and also 3.A element number is a totality number which has specifically two factors: 1 and itself. because that example, 7 is prime due to the fact that the only determinants of 7 are 1 and also 7.A composite number is a whole number with an ext than 2 factors. Because that example, 6 is a composite number due to the fact that it has actually 4 factors.**

We have the right to write any type of number together a product of several of its factors. A useful means to create a number is together a product of just prime numbers.

### Definition: prime Factorization

The prime factorization of a number is a product of element numbers which is same to the original number.

For example, 2×3×5 is the element factorization that 30 due to the fact that 2, 3, and also 5 are all prime and their product is 30.

### Example 1: finding the element Factorization of small Numbers

Which that the following is the element factorization that 18?

1822×92×3×33### Answer

To verify i beg your pardon product is the element factorization the 18, we must examine two things: that the product is same to 18 and also that the determinants in the product room all element numbers.

Since 18 is not prime, the is no the correct answer.Since no 2 nor 3 room equal to 18, they room not correct.Since 9 is no a prime number, 2×9 is not a prime factorization.We space left through 2×3×3. Both 2 and also 3 are prime, and the product is equal to 18. Hence, this is the prime factorization that 18.

The prime factorization of a number can be valuable when we require to uncover the highest common factors or lowest usual multiples of set of numbers, but we will certainly not comment on how to carry out this here. Instead, us will emphasis on two approaches to uncover the element factorization: variable trees and also dividing by primes. These methods take advantage of the truth that when you have actually written a number together a product that its factors, you can always exchange among the components for a product i m sorry that element is equal to.

For example, we recognize that 40=4×10. Yet we deserve to substitute 4 because that 2×2 or instead of 10 because that 2×5 in the equation and also the product on the right-hand side will still be same to 40. This provides us a method of writing numbers as a product that smaller and smaller factors. If we continue with our instance for 40, we will obtain to 40=2×2×2×5, at which allude we cannot malfunction any that the determinants into smaller sized numbers since they space all prime.

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The crucial to recognize prime components is the adhering to result.

### Result: factors of a Number

If