How to Find the Prime Factors of a Number: 14 Steps

2023 Author: Sybil Carroll | [email protected]. Last modified: 2023-05-21 06:08

Finding the prime factors of a number means breaking down that number down to the simplest building blocks. If you hate working with large numbers like 5733, learn how to transform it into 3 x 3 x 7 x 7 x 13. This type of problem is extremely important for cryptography and other techniques used to keep information secure. If you're not ready to create your own secure email system, you can at least start by learning how to factor to make fractions easier.

Steps

Part 1 of 2: Finding the Prime Factors

Step 1. Learn to factor

Factoring is the process of "breaking down" a number into smaller parts. These parts, or factors, are multiplied together to give the original number.

For example, to factor the number 18, you would break it down into 1 x 18 or 2 x 9 or 3 x 6

Step 2. Review the concept of "prime" numbers

A number is prime when it has only two factors: 1 and itself. The number 5, for example, is the product between 5 and 1. You cannot break it down into any other number. The goal of factoring is to keep breaking down until only prime numbers remain. This is especially useful when working with fractions, as it makes them easier to compare and use in equations.

Step 3. Start with a number

Pick any non-prime number greater than 3. There is no point in starting with a prime number since there is no way to factor it.

Example: this guide will look for prime factors of 24

Step 4. Factor the original number into any two numbers

Find two random numbers that multiply to give the starting number. You can use any two numbers, but if at least one of them is a prime it will be easier. A good strategy is to try dividing the number by two, then by 3, then by 5, and continue with higher and higher prime numbers until you find one that can perfectly divide the original number.

Example: If you don't know a factor of 24, start by trying to divide it by some small prime number. Divide by 2 to get 24 = 2 x 12. This doesn't end here, but at least it's a good start.
Since 2 is a prime number, it is always the easiest option to start factoring any even number.

Step 5. Begin to draw a factorization tree

Factoring trees are an excellent option for tracking a factoring problem. To start drawing a tree, simply draw two "branches" that separate downward from the original number. Write down the two factors at the end of these branches.

Example:
24
/\
2 12

Step 6. Factor the next line of numbers

Look at the two new numbers (the second line of the factoring tree). Are they both cousins? If any of them are not prime, continue factoring in the same way. Draw more branches on the tree and record the new factors on a third line.

Example: 12 is not a prime, so you must factor again. You can factor it as 12 = 2 x 6. Now add it to the factoring tree.
24
/\
2 12
/\
2 x 6

Step 7. Lower the prime numbers

If any of the factors is a prime number, move it down to the next line with its own simple "branch". There is no way to keep breaking it down, so from now on you just have to keep downloading it so as not to lose its trail.

Example: 2 is a prime number. Move the two down from the second line to the third.
24
/\
2 12
/ /\
2 2 6

Step 8. Continue factoring until only prime numbers remain

Review each line in the factoring tree when you finish doing it. If any of the numbers can be factored again, add a new line. You'll know you're done when only prime numbers remain.

Example: 6 is not a prime number, therefore it can be factored again. 2 is a prime number so you just have to lower it to the next row.
24
/\
2 12
/ /\
2 2 6
/ / /\
2 2 2 3

Step 9. Write the final line as prime factors

There will come a time when only prime numbers will remain. When this happens, it means you are done factoring. The result of the factoring is the last complete number line, written as a multiplication problem.

Check if you did your job well by multiplying all the numbers in the last line with each other. The result must be the original number.
Example: the final line of the factoring tree has only 2 and 3. They are both prime, which means you are done. Now you can write the prime factors of 24 as 24 = 2 x 2 x 2 x 3.
The order of the factors is not relevant. 2 x 3 x 2 x 2 is also a correct answer.

Step 10. Simplify using exponents (optional)

If you know how to write with exponents, you can make factoring easier to read. Remember, an exponent is a base number followed by a number ^high which indicates how many times the base is multiplied.

Example: in the factoring 2 x 2 x 2 x 3, how many times does the 2 appear? Since the answer is "three", you can simplify 2 x 2 x 2 as 2³. The simplified factorization then is 2³ x 3.

Part 2 of 2: Using Prime Factors

Step 1. Find the greatest common divisor of two numbers

The greatest common factor (GCF) of two numbers is the largest number that is a factor of both numbers. This is how you can find the GCF of 30 and 36 through their prime factors:

Find the prime factors of both numbers. The factorization of 30 is 2 x 3 x 5. The factorization of 36 is 2 x 2 x 3 x 3.
Find a number that appears between the factors of both numbers. Cross it off once on each list and write it on a new line. For example, 2 is in both lists so write 2 on a new line. Now you will have 30 = 2 x 3 x 5 and 36 = 2 x 2 x 3 x 3.
Repeat this step until there are no more factors in common. There is also a 3 in both lists so write it down on the new line, where you will now have

Step 2

Step 3.. Compare 30 = 2 x 3 x 5 with 36 = 2 x 2 x 3 x 3. There are no more numbers in common.
To find the GCF, multiply all the shared factors. In this example you only have 2 and 3, so the GCF is 2 x 3 =

Step 6.. This is the largest number that is a factor of both 30 and 36.

Step 2. Simplify fractions using the GCF

Use the greatest common divisor whenever you suspect that a fraction is not expressed in its simplest form. Find the GCF of the numerator and denominator using the process explained above. Once you find it, divide both parts of the fraction by the GCF. The answer will be the same fraction but in its simplest (irreducible) form. The answer will be the same fraction in simplest form.

For example, simplify the fraction ³⁰/₃₆. You already found out that the GCF is 6, so now divide both the numerator and denominator by 6:
30 ÷ 6 = 5
36 ÷ 6 = 6
³⁰/₃₆ = ⁵/₆

Step 3. Find the least common multiple of two numbers

The least common multiple (m.c.m.) of two numbers is the smallest number that has those first two numbers as factors. For example, the m.c.m. 2 and 3 is 6 because 6 has 2 and 3 as factors. Here is an example of how to find the m.c.m. from a factorization:

Start by factoring the two numbers. For example, the factorization of 126 is 2 x 3 x 3 x 7. The factorization of 84 is 2 x 2 x 3 x 7.
Compare the number of times each unique factor appears in the two lists. Choose a list where it appears the most times and circle each instance. For example, 2 appears once among the factors of 126, but twice in the list of 84. Circle the 2 x 2 in the second list.
Repeat this step for each unique factor. For example, 3 appears most often in the first list, so enclose 3 x 3 on that list. 7 only appears once in each list, so enclose only one

Step 7. (when there is a tie you can choose the list you want).
Multiply all the numbers you circled to get the lcm. In this example, the least common multiple of 126 and 84 is 2 x 2 x 3 x 3 x 7 = 252. This is the smallest number that has 126 and 84 as factors.

Step 4. Use the least common multiple to add fractions

In order to add two fractions, their denominators must be equal. Find the least common multiple of both denominators. Multiply each fraction so that the new denominator becomes m.c.m. Once the two fractions are expressed in this way, you can proceed to add them.

For example, imagine you want to solve ¹/₆ + ⁴/₂₁.
Using the above method, you have to find the m.c.m of 6 and 21. The answer is 42.
Converts ¹/₆ in a fraction with 42 as the denominator. To do this, solve 42 ÷ 6 = 7. Multiply ¹/₆ x ⁷/₇ = ⁷/₄₂.
To transform ⁴/₂₁ In a fraction with 42 as the denominator, solve 42 ÷ 21 = 2. Multiply ⁴/₂₁ x ²/₂ = ⁸/₄₂.
Now that the two fractions are expressed with the same denominator, you can easily add them: ⁷/₄₂ + ⁸/₄₂ = ¹⁵/₄₂.

Practical problems

Try to solve these problems by yourself. When you think you have the correct answer, select the blank text to make it visible and you will see the correct answer. The last problems are more difficult.
Find the prime factors of 16: 2 x 2 x 2 x 2
Write the answer using exponents: 2⁴
Find the prime factors of 45: 3 x 3 x 5
Write the answer using exponents: 3² x 5
Find the prime factors of 34: 2 x 17
Find the prime factors of 154: 2 x 7 x 11
Find the prime factors of 8 and 40, then find the greatest common factor of both: The factorization of 8 is 2 x 2 x 2 x 2. The factorization of 40 is 2 x 2 x 2 x 5. Its GCF is 2 x 2 x 2 = 6.
Find the prime factors of 18 and 52, then find the least common multiple of both: The factorization of 18 is 2 x 3 x 3. The factorization of 52 is 2 x 2 x 13. Your m.c.m. is 2 x 2 x 3 x 3 x 13 = 468.

Advice

All numbers have a unique factorization. No matter what factors you choose along the way, you will always end up arriving at this unique result. This is what is known as the "fundamental theorem of arithmetic."
Instead of moving the prime numbers down to the next line in the factoring tree, you can leave them where they are and circle them. Once you finish factoring, the prime factors will be all those numbers that are circled.
Always check your work. You can make simple mistakes without realizing it.
Beware of misleading questions. If you are asked to find the prime factors of a number that is already prime, you have nothing to do. The only prime factor of 17 is 17. There is no way to break down that number further.
You can find the greatest common divisor or the least common multiple of three or more numbers.