Methodprime factorizationis used for "demolishor express a specific number asproduct of prime numbers. When a prime occurs more than once after factoring, it is usually expressed in exponential form to make it appear more compact.
Otherwise, we have a long list of primes that are multiplied together. In this lesson, I have prepared eight (8) working examples to explain the factorization process using prime numbers.
Note:When I say "numbers" here, I mean themnatural numbers.
Definition of a prime number and examples and non-examples of prime numbers
Let's start by defining what a prime number is. Study carefully both examples and counterexamples of prime numbers.
What is a prime number?
A prime [latex]p[/latex] is an integer greater than 1 that is only divisible by 1 and itself. In other words, the prime number has exactlytwo factors, namely: 1 and yourself.
Examples of prime numbers:
- Numer2is a prime number because it is only divisible by 1 and 2 (by itself).
- Numer17is a prime number because it has exactly two divisors, namely 1 and 17 (itself)
- Numer31is a prime number because it can only be divided by two numbers, namely 1 and 31 (alone)
Examples of non-prime numbers:
- Numer10is not a prime number because it is divisible by 1, 5, and 10 (by itself) and therefore has more than two divisors
- Numer27is not prime because it is divisible by more than two factors except 1 and itself, which contains 3 and 9
- Numer49is not a prime number because, except for 1 and 49 (alone), it is divisible by 7, so it has more than two divisors
Now I will explain the general steps involved in factoring a given positive integer.
How to do prime number factorization
Step 1:List at least the first few primes in ascending order. I'll stop at 19 because that's plenty for the numbers in this tutorial that we'll be covering.
2, 3, 5, 7, 11, 13, 17, 19…
step 2:For any number, check whether it is divisible by the smallest prime number, which is 2. If the prime number 2 divides the given number evenly, express them as factors:
step 3:Check again if the second number that comes out is divisible by 2. If so, continue until the new number is no longer divisible by 2. Two things can happen here:
- When you divide 2 multiple times, you get a prime number. Go to the last step. You're almost ready!
- When you divide 2 many times, you get a composite number (not prime), but one that cannot be divided by 2. Go to step 4.
step 4:Go to the next larger primes, such as 3, 5, 7, etc. If needed, to see if the number we left off in the previous step can be further divided evenly. This iterative process of dividing by primes in ascending order will eventually give us the final prime factor.
last step:At this point, we should have a long list of primes that are multiplied together. We just need to represent our final answer as a product of exponential terms with prime bases. That's it! A quick example might look like this...
The exponent of each prime tells you how often that prime occurs as a factor.
Examples of performing prime factorization using a prime factor tree and inverse division
Example 1:Find prime factorization40and express it in exponential notation.
I'll start by listing the first few primes in ascending order. The goal is to continue dividing the given number, starting with the smallest number, by the corresponding prime number until the last quotient is also a prime number.
There are two common ways to perform prime factorization. The first is calledprime factors treeand the other is calledDivision upside down. I will also show you two ways to prime the number 40.
prime factors tree
- First, divide the given number by the smallest prime number, which is 2.
- Factors with the above number are divided into "branches", as indicated by the line segments.
- we can share40and its quotient by the prime number 2 three times, which means that this prime has an exponent of 3 when factored.
- The last quotient after multiple divisions of 2 is a prime number, which is 5.
- Reaching prime as the last quotient in the process shows we're done!
Division upside down
- Now you know why it's called upside-down splittingThe split symbol is literally inverted.
- I start dividing the given number by the smallest prime number, which is 2. If this prime number divides the number evenly, I put the quotient below. Continue as needed.
- Notice that we can keep dividing prime 2 until we reach prime 5 as the final integer quotient (bottom).
- Express the final factorization as the product of base 1 exponents in exponential notation.
Example 2:Find prime factorization32and express it in exponential notation.
It's an even number and therefore divisible by the prime number 2. So without hesitation I start using them as my initial divisor of choice.
prime factors tree
- First, divide the given number by the smallest prime number, which is 2.
- Factors with the above number are divided into "branches", as indicated by the line segments.
- After dividing by 2 many times, we also arrive at the final factor of 2. That's it!
- Since prime 2 occurs as a factor five times, the final answer is [latex]32 = {2^5}[/latex].
Division upside down
- I start dividing the given number by the smallest prime number, which is 2. If this prime number divides the number evenly, I put the quotient below. Continue as needed.
- It's great! We only used one prime to figure it all out.
- The prime number 2 occurs five times as a factor. So our answer is simply [latex]32 = {2^5}[/latex].
Example 3:Find prime factorization147and express it in exponential notation.
First, I realize that 147 is an odd number and therefore not divisible by 2. Continue with the next larger prime, which is 3.
Not sure if you've come across the "nice" divisibility rule for the number 3. It states that if the sum of the digits of a number is divisible by 3, then the primitive is also divisible by 3.
We have 147, which is the sum of its digits147 = 1 + 4 + 7 = 12which is divisible by 3. This means that 147 must also be divisible by3.
You don't need to display the prime factorization of a number every time you use these two methods. Just choose the option that is easy or convenient for you. In this exercise, I will use the factor tree method.
prime factors tree
- Since the prime number 2 cannot divide 147 evenly, check the next larger prime number, which is 3. Yes! This should work.
- After dividing 147 by 3 I get 49 which is obviously a perfect square because [latex] 49 = {7^2}[/latex]
- The next logical prime that you can use as a divisor is 7.
Example 4:Find prime factorization540and express it in exponential notation.
We know that every even number is always divisible by 2. So I would start by dividing 540 by the prime number 2. Let me use upside-down division to factor this number.
Division upside down
- The given number 540 is even, so it can be divided by 2. Here we do two consecutive divisions, using the prime number 2 as the divisor.
- We get 135 as a partial divisor that is divisible by the next larger prime, which is 3!
- Repeat division by 3 and you get the last factor of 5 which is a prime number. That's it! Simply collect the various prime numbers and assign the appropriate powers or exponents to represent the final answer.
Example 5:Find prime factorization945and express it in exponential notation.
The prime factorization can also be done in the simplest way, i.e. by subtracting the number horizontally and along the line. Just make sure you always start with the lowest prime number and move on to the next larger one if necessary to break it up until you end up with the final prime factor. Some manuals do this to save space. It's nice to add this method to your math "toolbox".
Solution:
It hasfive main factorswhere there are three of them (3).recognizable, namely: 3, 5 and 7.
Example 6:Find prime factorization1320and express it in exponential notation.
Solution:
Example 7:Find prime factorization2025and express it in exponential notation.
ATTENTION: The given number is odd, so the prime number 2 cannot divide it evenly. Start with 3.
Solution:
Example 8:Find prime factorization432and express it in exponential notation.
NOTE: I used a slightly different method here. I call it"Composite stays inside, Prime outside"I mean the parenthesis that indicates where the composite or prime number is.
Solution:
The goal is to keep the unfolded primes out of the brackets while forcing the composite numbers to stay in the brackets. After all, we should have all the primes and the parentheses will be gone!
You may be interested in:
the principle of arithmetic
List of prime factorizations of integers from 2 to 200
List of prime factorizations of integers from 201 to 400
List of prime factorizations of integers from 401 to 600
List of prime factorizations of integers from 601 to 800
FAQs
What is the prime factorization of 2 into 7 into 11 into 13 into 17 21? ›
The prime factorization of given , 2 × 7 × 11 × 13 × 17 + 21 is 3 × 7 × 11 × 13 × 17 + 3 × 7.
Does every integer have a prime factorization? ›Definitions: The prime factors of a positive integer are the prime numbers that divide that integer exactly. The process of finding these numbers is called integer factorization, or prime factorization. The fundamental theorem of arithmetic says that every positive integer has a unique prime factorization.
What is the prime factorization of 24 and 32? ›Prime factorization of 24 and 32 is (2 × 2 × 2 × 3) and (2 × 2 × 2 × 2 × 2) respectively. As visible, 24 and 32 have common prime factors. Hence, the HCF of 24 and 32 is 2 × 2 × 2 = 8.
What is the prime factorization of the number 32? ›| Factors of 32 | Factor pairs of 32 | Prime Factorization of 32 |
|---|---|---|
| 1, 2, 4, 8, 16 and 32 | (1, 32), (2, 16), (4, 8) and (16, 2) | 2 x 2 x 2 x 2 x 2 |
What are the prime numbers from 1 to 100? The prime numbers from 1 to 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
What is the prime factor of 2 into 7 into 11 into 23 23? ›The prime factor of 2 x 7 x 11 x 17 x 23 + 23 is 23.
What is the prime factorization of 1212? ›So, the prime factorization of 1212 can be written as 22 × 31 × 1011 where 2, 3, 101 are prime.
What is an example of factorization of integers? ›x5+5×x4+10×x3+10×x2+5×x+1=(x+1)5. As shown in the above example, when dealing with lots of numbers, it is sometimes helpful to substitute a number that seems to follow a certain pattern or have a repetitive structure, with a variable and simplify it as we would any other expression.
Can an integer be prime? ›By the usual definition of prime for integers, negative integers can not be prime. By this definition, primes are integers greater than one with no positive divisors besides one and itself.
How many primes are in integers? ›There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled.
What is the prime factorization of 18 and 32? ›
Prime factorization of 18 and 32 is (2 × 3 × 3) and (2 × 2 × 2 × 2 × 2) respectively. As visible, 18 and 32 have only one common prime factor i.e. 2.
What is the prime factorization method of 18 and 24? ›Prime factorization of 18 and 24 is (2 × 3 × 3) and (2 × 2 × 2 × 3) respectively. As visible, 18 and 24 have common prime factors.
What is the prime factorization of 18 24 and 30? ›Prime factorization of 18, 24, and 30 is (2 × 3 × 3) = 21 × 32, (2 × 2 × 2 × 3) = 23 × 31, and (2 × 3 × 5) = 21 × 31 × 51 respectively.
What is the prime factorization of 240? ›The prime factorization of 240 is 2 × 2 × 2 × 2 × 3 × 5 or 24 × 3 × 5.
What is the prime factorization of the number 35? ›Hence, the prime factorization of 35 is 5 x 7.
What is the prime factorization of 45 and 32? ›Prime factorization of 32 and 45 is (2 × 2 × 2 × 2 × 2) and (3 × 3 × 5) respectively.
What is the prime factorization of 24? ›The prime factorisation of 24 gives 2 x 2 x 2 x 3 = 23 x 3, where 2 and 3 are the prime factors of 24. The sum of factors of 24 is 60.
What is the prime factorization of the number 46? ›Therefore, the positive factors of 46 are: 1, 2, 23, and 46. Also, the negative pair factors of 46 are (-1, -46), (-2, -23), (-23, -2), and (-46, -1).
What is the prime factorization of the number 30? ›Its Prime Factors are 1, 2, 3, 5, 6, 10, 15, 30 and (1, 30), (2, 15), (3, 10) and (5, 6) are Pair Factors. What are Factors of 30?
Is there a pattern to find prime numbers? ›A clear rule determines exactly what makes a prime: it's a whole number that can't be exactly divided by anything except 1 and itself. But there's no discernable pattern in the occurrence of the primes.
Why is finding prime numbers difficult? ›
"Mathematicians don't, generally speaking, go around looking for prime numbers. The main reason is that we know there's infinitely many prime numbers, so you're never going to get to the end of the list," Solomon says.
What is the pattern rule for 0 1 3 6 10? ›These numbers are in a sequence of 1, 3, 6, 10, 15, 21, 28, 36, 45, and so on. The numbers in the triangular pattern are represented by dots. The sum of the previous number and the order of succeeding number results in the sequence of triangular numbers.
What is the next number in the sequence 2 3 4 7 6 11 8 15 10 * 12 13 17 19? ›Detailed Solution
Hence, '23' is the correct answer.
Apart from 2 and 5, all prime numbers have to end in 1, 3, 7 or 9 so that they can't be divided by 2 or 5.
What is the total number of prime factor 7 2 24 4 5 21? ›∴ The total number of prime factors of the given number is three.
Are 2 and 9 the prime factors of 18? ›It has a total of 6 factors of which 18 is the biggest factor and the positive factors of 18 are 1, 2, 3, 6, 9, and 18. The Pair Factors of 18 are (1, 18), (2, 9), and (3, 6) and its Prime Factors are 1, 2, 3, 6, 9, 18.
How many prime factors does 9 * 2 24 have? ›So, there are 75 prime factors in .
What is the prime factorization 12 15 9? ›Prime factorization of 9, 12, 15: 9 = 3. 12 = 22 × 3. 15 = 31 × 5.
What is the prime factorization of 1234? ›The Prime Factors and Pair Factors of 1234 are 2 × 617 and (1, 1234), (2, 617) respectively.
What are the 3 rules of factorization? ›- Law of Sine.
- Law of Cosines.
- Law of Tangent.
What is the integer factorization problem? ›
The integer factorization problem is defined as follows: given a composite number N, find two integers x and y such that x · y = N. Factoring is an important problem because if it can be done efficiently, then it can be shown that RSA encryption is insecure.
What are the 4 types of factorization? ›- Greatest common factor.
- Grouping.
- Difference in two squares.
- Perfect square trinomial pattern.
A fundamental theorem in number theory states that every integer n ≥ 2 can be factored into a product of prime powers. This factorisation is unique in the sense that any two such factorisations differ only in the order in which the primes are written. For example: 12 = 22*3.
Is the integer 2 a prime? ›Is 2 a prime number? Yes, because its only factors are 1 and itself.
Are all prime integers odd? ›Looking at this short list of prime numbers can already reveal a few interesting observations. First, except for the number 2, all prime numbers are odd, since an even number is divisible by 2, which makes it composite.
What is the prime factorization of 84? ›The prime factorization of 84 gives us 2 × 2 × 3 × 7.
What is the prime factorization of 30? ›Its Prime Factors are 1, 2, 3, 5, 6, 10, 15, 30 and (1, 30), (2, 15), (3, 10) and (5, 6) are Pair Factors.
How do you find the prime factorization of 360? ›There are overall 24 factors of 360, of which 2, 3 and 5 are its prime factors. The Prime Factorization of 360 is 23 × 32 × 5.
How do you find the prime factorization of 414? ›So, the prime factorization of 414 can be written as 21 × 32 × 231 where 2, 3, 23 are prime.
How do you find the prime factorization of the greatest 4 digit number? ›Therefore, 9999 is the largest 4-digit number and can be expressed as 3 × 3 × 11 × 101.
What is the prime factorization of 128? ›
The number 128 is a composite number, that is, it has more than two numbers as factors. To find the prime factors, first we will divide the number 128 by its smallest prime factor, that is, 2. So, the prime factorization of 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 or 27.
What is the prime factorization of 120? ›So the prime factorization of 120 is 23×3×5.
What is the prime factorization of 99? ›Hence, the prime factorization of 99 is 3 x 3 x 11 = 32 x 11 where 3 and 11 are prime numbers.
What is the prime factorization of 96? ›Prime Factorization refers to finding which prime numbers multiply together to form the original number. The prime factors of 96 thus obtained are written as 96 = 2 × 2 × 2 × 2 × 2 × 3 = 25 × 31, where 2 and 3 are the prime numbers.
What is the prime factorization of 87? ›Solution : Factors of 87 = 1, 3, 29 and 87. The prime factors of 87 are 3 and 29.
What is the prime factorization of 460? ›So, the prime factorization of 460 can be written as 22 × 51 × 231 where 2, 5, 23 are prime.
What is the prime factorization of 98? ›So, the prime factorization of 98 = 2 × 7 × 7 or 2 × 72. This means that 2 and 7 are the prime factors of the number 98.
What is the prime factorization of 200? ›So, the prime factorization of 200 is 200 = 2 × 2 × 2 × 5 × 5 and the prime factors of 200 are 2 and 5.
What is the prime factorization of 500? ›The prime factorization of 500 is: 2 x 2 x 5 x 5 x 5.