How to factor numbers

Here are the steps required for factoring a trinomial when the leading coefficient is not 1:

Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power.
Step 2 : Decide if the three terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.
Step 3 : Multiply the leading coefficient and the constant, that is multiply the first and last numbers together.
Step 4 : List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to x.
Step 5 : After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to x and also multiply to equal the number found in Step 3.
Step 6 : Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5.
Step 7 : Now that the problem is written with four terms, you can factor by grouping.

Example 1 – Factor:

Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power. In this case, the problem is in the correct order.
Step 2: Decide if the three terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer. In this case, the three terms only have a 1 in common which is of no help.
Step 3: Multiply the leading coefficient and the constant, that is multiply the first and last numbers together. In this case, you should multiply 6 and –2.
Step 4: List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to x. In this case, the numbers 3 and 4 can combine to equal 1.
Step 5: After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to x and also multiply to equal the number found in Step 3. In this case, –3 and +4 combine to equal +1 and –3 times +4 is –12. How to Factor Numbers
Step 6: Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5. How to Factor Numbers
Step 7: Now that the problem is written with four terms, you can factor by grouping. How to Factor Numbers

Example 2 – Factor: How to Factor Numbers

Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power. In this case, the problem is in the correct order. How to Factor Numbers
Step 2: Decide if the three terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer. In this case, the three terms only have a 1 in common which is of no help. How to Factor Numbers
Step 3: Multiply the leading coefficient and the constant, that is multiply the first and last numbers together. In this case, you should multiply 12 and 15. How to Factor Numbers
Step 4: List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to x. In this case, the numbers 9 and 20 can combine to equal 29.
Step 5: After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to x and also multiply to equal the number found in Step 3. In this case, –9 and –20 combine to equal –29 and –9 times –20 is 180. How to Factor Numbers
Step 6: Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5. How to Factor Numbers
Step 7: Now that the problem is written with four terms, you can factor by grouping. How to Factor Numbers

Click Here for Practice Problems

Example 3 – Factor:

Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power. In this case, the problem needs to be rewritten as:
Step 2: Decide if the three terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer. In this case, the three terms have a 2 in common, which leaves:
Step 3: Multiply the leading coefficient and the constant, that is multiply the first and last numbers together. In this case, you should multiply 3 and –8.
Step 4: List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to x. In this case, the numbers 4 and 6 can combine to equal 2.
Step 5: After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to x and also multiply to equal the number found in Step 3. In this case, +4 and –6 combine to equal –2 and +4 times –6 is –24.
Step 6: Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5. Do not forget to include 2 (the GCF) as part of your answer.
Step 7: Now that the problem is written with four terms, you can factor by grouping. Do not forget to include 2 (the GCF) as part of your final answer.

Click Here for Practice Problems

Example 4 – Factor:

Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power. In this case, the problem is in the correct order.
Step 2: Decide if the three terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer. In this case, the three terms have a 3x in common, which leaves:
Step 3: Multiply the leading coefficient and the constant, that is multiply the first and last numbers together. In this case, you should multiply 12 and 2.
Step 4: List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to x. In this case, the numbers 3 and 8 can combine to equal 11.
Step 5: After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to x and also multiply to equal the number found in Step 3. In this case, +3 and +8 combine to equal +11 and +3 times +8 is 24.
Step 6: Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5. Do not forget to include 3x (the GCF) as part of your answer.
Step 7: Now that the problem is written with four terms, you can factor by grouping. Do not forget to include 2 (the GCF) as part of your final answer.

Click Here for Practice Problems

Note: In each example above you could not use the shortcut because the leading coefficient was not a 1.

How to Factor Out Numbers

Factoring is the opposite of distributing. When distributing, you multiply a series of terms by a common factor. When factoring, you seek to find what a series of terms have in common and then take it away, dividing the common factor out from each term.

Think of each term as a numerator and then find the same denominator for each. By factoring out, the factor is put outside the parentheses or brackets and all the results of the divisions are left inside.

The proper way to factor an expression is to write the prime factorization of each of the numbers and look for the greatest common factor.

A more practical and quicker way is to look for the largest factor that you can easily recognize. Factor it out and then see if the numbers within the parentheses need to be factored again.

Repeat the division until the terms within the parentheses are relatively prime.

Example: Follow these steps to factor out the expression

How to Factor Numbers

  1. Determine a common factor.

    A common factor is 2.

  2. Divide each term by the common factor and write the results of the division in parentheses, with the factor out in front.

    How to Factor Numbers

  3. Determine whether you can factor out any other terms.

    The terms left in the parentheses are still too large. They all still a common factor of 4. Factoring out 4, you get:

    How to Factor Numbers

  4. Simplify the answer.

    If you factor out a 4 after factoring out the 2, then the product of 4 and 2 (which is 8), is the total amount you factored out. The final answer is

    How to Factor Numbers

Prime Factorization

Prime FactorizationFactor Trees

Computer Security

As mentioned in Session 14, prime numbers are important in computer security such as with Public Key Cryptography. A concern in computer security is the ability to factor large numbers.

One of the reasons that computers can maintain security is that many large numbers are difficult to factor into products of primes. Should someone find a method to easily factor any large number or to test if it is prime, computer security would no longer be secure.

At one time, RSA Laboratories offered significant monetary awards for challenge problems involving factoring large numbers.

  • Integer factorization – Wikipedia, the free encyclopedia
  • RSA Factoring Challenge
  • Can prime numbers be thought of as the building blocks of natural numbers?

We know that 42 = 2 × 3 × 7. Is there another way to represent 42 as a product of primes?

The only way to write 42 as the product of primes (except to change the order of the factors) is 2 × 3 × 7. We call 2 × 3 × 7 the prime factorization of 42.

It turns out that every counting number (natural number) has a unique prime factorization, different from any other counting number. This fact is called the Fundamental Theorem of Arithmetic.

Fundamental theorem of arithmetic – Wikipedia, the free …

In order to maintain this property of unique prime factorizations, it is necessary that the number one, 1, be categorized as neither prime nor composite. Otherwise a prime factorization could have any number of factors of 1, and the factorization would no longer be unique.

Prime factorizations can help us with divisibility, simplifying fractions, and finding common denominators for fractions.  

Factor Trees

One method for producing the prime factorization of a natural number is to use what is called a factor tree.  

The first step in making a factor tree is to find a pair of factors whose product is the number that we are factoring. These two factors are the first branching in the factor tree. There are often several different pairs of factors that we could choose to begin the process.

The choice does not matter; we may begin with any two factors. We repeat the process with each factor until each branch of the tree ends in a prime. Then the prime factorization is complete.

The Fundamental Theorem of Arithmetic guarantees that all prime factorizations of the same number will result in the same, unique prime factorization for the number.

Example: We show two of the ways of constructing a factor tree for 24.

How to Factor Numbers

Continue factoring each tree until complete.

How to Factor Numbers

Note that each tree ends with the unique prime factorization of 24 = 2 · 2 · 2 · 3 = 23 · 3.

There are two different styles for writing down the factor tree of a natural number. In the first style, as soon as we obtain a prime number in one of the branches, we circle it and then do not work on that branch any more.

If a number at the end of a branch is still not prime (a composite), we find two factors for that value. Continue this process until the value at the end of each branch is a circled prime number.

The prime factorization is the product of the circled primes.

Example:

How to Factor Numbers

So the prime factorization of 24 is 24 = 2 · 2 · 2 · 3 = 23 · 3.

A good way to check the result is to multiply it out and make sure the product is 24.

For the other style of factor tree, we maintain the product of the original value at each level of the factor tree by extending the branch (bringing down) for any prime obtained on the way to getting all of the branches to end in prime numbers.The following example shows this style and also how we may start with a different pair of factors and still come out with the same prime factorization for the natural number.

Example:

Factor – Definition with Examples

Multiplying two whole numbers gives a product. The numbers that we multiply are the factors of the product.  

How to Factor Numbers

  • Example:  3 × 5 = 15 therefore, 3 and 5 are the factors of 15.
  • This also means:
  • A factor divides a number completely without leaving any remainder.

How to Factor Numbers

For example:  30 ÷ 6 = 5, and there is no remainder. So we can say that 5 and 6 are the factors of 30.

In the given example, we can further break up or simplify the number 6 into its factors, that is, 2 and 3. In other words, when we multiply 5, 2 and 3, we still get 30. Therefore, the factors of 30 are 5, 2, and 3. Also, 5 × 2 = 10. So, 10 is also a factor of 30. Similarly, 5 × 3 = 15. So, 15 is also a factor of 30. Finally, the factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30

  • The number 1 is the smallest factor of every number.
  • Every number will have a minimum of two factors, 1 and the number itself.
  • A number that has only two factors, 1 and the number itself, is called a prime number.

Prime factorization

  • When we write a number as a product of all its prime factors, it is called prime factorization.
  • Every number in prime factorization is a prime number.
  • Sometimes to write the prime factors of a number we might have to repeat a number.

 How to Factor Numbers

  Fun Facts

  • Factors are always whole numbers or integers and never decimals or fractions.
  • All even numbers will have number 2 as their factor.
  • All numbers that end with 5 will have 5 as their factor.
  • All numbers greater than 0 and ending with a 0 will have 2, 5, and 10 as their factors.
  • Algebraic expressions are often solved or simplified through factoring.

All about Factors of a number | Factoring Numbers – Hitbullseye

Factors of a number is an important sub-topic from number systems. In this article, we will discuss the fundamentals of factors of a number. Almost every competitive examination has 2-3 medium to difficult level questions based on factors of a number. Taking this into consideration, we will discuss the advanced application of this topic, to give you an edge over other candidates.

How to Factor Numbers

Factors of a number N refers to all the numbers which divide N completely. These are also called divisors of a number.

  • Basic formula related to factors of a number:
  • These are certain basic formulas pertaining to factors of a number N, such that,
  • N= paqbrc
  • Where, p, q and r are prime factors of the number n.
  • a, b and c are non-negative powers/ exponents
  • Number of factors of N = (a+1)(b+1)(c+1)
  • Product of factors of N = N No. of factors/2
  • Sum of factors: ( p0+p1+…+pa) ( q0+ q1+….+qb) (r0+r1+…+rc)/ (pa-1)(qb-1)(rc-1)

Solved questions on Factors of a number:

Example 1: Consider the number 120. Find the following for n

  1. Sum of factors
  2. Number of factors
  3. Product of factors

Solution: The prime factorization of 120 is 23x31x51. By applying the formulae,

  • Sum of factors = [(20+21+22+23)(30+31)(50+51)]/ [(2-1) (3-1)(5-1)] = 45
  • Number of factors = (3+1)(1+1)(1+1) = 16
  • Product of factors = 120(16/2) = 1208

Example 2: Find the following for the number 84 :-

  1. Number of odd factors
  2. Number of even factors

Solution: By the prime factorization of 84, 84= 22 × 31 × 71
Total number of factors = (2+1)(1+1)(1+1) = 12

  1. Number of odd factors will be all possible combinations of powers of 3 and 5 (excluding any power of 2) . Hence number of odd factors = (1+1)(1+1) = 4 By manually checking, these factors are 1, 3, 7 and 21.
  2. Number of even factors = total no. of factors – no. of even factors

    = 12 – 4 = 8

Know where you stand in Factors. Take this test now.

Advanced Concepts of Factors of a number

Example 3: Let N= 315×743. How many factors of N2 are less than N but do not divide N completely?

Solution: Let N= 6, then, N2 = 36.

Factors Of a Number – Formula to find Total Number of factors

Factors of a number are defined as numbers or algebraic expressions that divide a given number/expression evenly.  We can also say, factors are the numbers which are multiplied to get another number.

For example, 1, 3 and 9 are the factors of 9, because 1 × 9 = 9 and 3 × 3 = 9. Here, the concepts of factors are explained which will help to understand how to find the factors and know the prime factors of some common digits.

Here we will discuss finding factors, formulas to find the number of factors, product and sum of factors.

Definition of Factors of a Number

The factors of a number are defined as the number which can be multiplied to get the original number. By multiplying two factors of a number, a product is obtained which is equal to the original number. It should be noted that factors of any number can be either positive or negative.

For example, in the case of 6, the factors can be 2 and 3 as 2 × 3 gives 6. Here, 2 and 3 are factors while 6 is the product. The other factors of 6 are 1 and 6, etc.

We can also consider -1, -2, -3 and -6 as the factors of 6, because when we multiply any two negative numbers, it results in positive number, such as;

  • -1 × -6 = 6
  • -2 × -3 = 6
  • Hence, the factors of 6 in total are 1,2,3,6,-1,-2,-3 and -6.
  • But normally, the factors are considered to be only positive numbers.
  • Point to remember: Fractions could not be considered as factors for any number.

Factors Formulas

There are basically three types of formulas considered for factors. They are:

  • Number of Factors
  • Product of Factors
  • Sum of Factors
  1. Let us assume N is a natural number, for which we need to find the factors. If we convert N into the product of prime numbers by prime factorisation method, we can represent it as;
  2. N = Xa × Yb × Zc
  3. where X, Y and Z are the prime numbers and a, b and c are their respective powers.
  4. Now, the formula for the total number of factors for a given number is given by;
  • Total Number of Factors for N = (a+1) (b+1) (c+1)

The formula for the sum of all factors is given by;

  • Sum of factors of N = [(Xa+1-1)/X-1] × [(Yb+1-1)/Y-1] × [(Zc+1-1)/Z-1]

The formula for the product of all factors is given by;

  • Product of factors of N = NTotal No. of Factors/2
  • Example: Find the total number of factors of 90 along with sum and product of all factors.
  • Solution: Write the prime factorisation of 90 first.
  • 90 = 2 × 45 = 2 × 3 × 15 = 2 × 3 × 3 × 5
  • 90 = 21 × 32 × 51
  • Here, X = 2, Y = 3, Z =5 and a = 1, b = 2, c = 1
  • Therefore, total number of factors of 90 = (a +1)(b+1)(c+1) = (1+1)(2+1)(1+1) = 2 × 3 × 2 = 12
  • Sum of factors of 90 = [(21+1-1)/2-1] × [(32+1-1)/3-1] × [(51+1-1)/5-1] = 3/1 × 26/2 × 24/4 = 3 × 13 × 6 = 234
  • Product of factors of 90 = 90Total factors of 90/2 = 9012/2 = 906

How to Find Factors of a Number?

Knowing how to calculate factors of a number is extremely crucial in maths. The steps to find the factors of a number are given below in a very easy to understand way. An example is taken to make the explanation easier.

  • Step 1: Choose a number (say, 16)
  • Step 2: Write the common factors of 16 which will include (16 × 1), (-16 × -1), (8 × 2), (-8 × -2), (4 × 4), and (-4 × -4).
  • Step 3: Further factor the factors until a prime number is reached. In this case, 8 can be factored further.
  • Step 4: Write down all the factors again. The (8 × 2) will now become (4 × 2 × 2).
  • Step 5: Write all the unique number that is obtained.

So, the factors of 16 will be 1, 2, 4, 8, 16, – 1, – 2, – 4, – 8, and – 16. Here, the positive factors of 16 are only 1, 2, 4, 8, and 16.

  1. Another Example:
  2. Consider the number as 80.
  3. 80 = 10 × 4
  4. = (5 × 2) × 8
  5. =(5 × 2) × (4 × 2)
  6. = (5 × 2) × (2 × 2 × 2)

Now, the factors of 80 will all the combination from 5 × 2 × 2 × 2 × 2 and 1 itself (as 1 × 80 = 80). So, the positive factors of 80 are 1, 2, 4, 5, 8, 10, 16, 20, 40, and 80. It should be noted that there will also be negative factors whose count have to be even.

How to Calculate Factors of Large Numbers?

To calculate the factors of large numbers, divide the numbers with the least prime number i.e. 2. If the number is not divisible by 2, move to the next prime numbers i.e. 3 and so on until 1 is reached. Below is an example to find the factors of a large number.

Example: 1420

Steps Prime Factors Product
Step 1: Divide by 2 2 710
Step 2: Again Divide by 2 2 355
Step 3: Divide by 5 71

In step 3, a prime number is obtained as a product and so, the process is stopped. The factors will be all the multiples of 1, 2, 2, 5, 71, 355, 710. Now, the positive factors of 1420 will be 1, 2, 4, 5, 10, 20, 71, 142, 284, 355, 710, and 1420.

In the same case, if only prime factors are considered, it is called the prime factorization of that number. In this way, it is easy to factor a number and know its factors and prime factors.

Factors of Some Common Numbers List

Below is a list of common numbers with their factors and prime factors. Each of these links will include the process of factoring of any number along with all the factors including the prime factors of that number.

Keep visiting BYJU’S to learn more such maths concepts in an easy and effective way. Also, register now to get complete assistance for maths preparation from various video lessons and other study materials.

How to Find the Prime Factorization of a Number – Video & Lesson Transcript

When you are trying to come to a conclusion about a problem, you often say that there are many 'factors' to consider.

This means that there are many parts that make up the whole problem of what you are trying to decide.

If the decision is where to go for dinner, the factors involved in that decision might be price, how far away the restaurant is, and how well you will enjoy the food.

Numbers also have factors, the parts that make up the whole number. The factors of a number are the numbers that, when multiplied together, make up the original number.

For example, factors of 8 could be 2 and 4 because 2 * 4 is 8.

And factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24, because 1 * 24 is 24, 2 * 12 is 24, 3 * 8 is 24 and so is 4 * 6. So all of these numbers are said to be factors of 24.

A prime number is any number that is only divisible by itself and 1. Some examples of prime numbers include 2, 5 and 17. Numbers such as 15 or 21 are not prime, because they are divisible by more than just themselves and 1.

Prime Factorization

To factor a number is to break that number down into smaller parts. To find the prime factorization of a number, you need to break that number down to its prime factors.

How to Determine the Prime Factorization of a Number

How to Factor a Number

  1. 1

    Write your number above a 2-column table. While it's usually fairly easy to factor small integers, larger numbers can be daunting.

    Most of us would be hard-pressed to break a 4 or 5-digit number into its prime factors using nothing but mental math. Luckily, using a table, the process becomes much easier.

    Write your number above a t-shaped table with two columns – you'll use this table to keep track of your growing list of factors.[4]

    • For the purpose of our example, let's choose a 4-digit number to factor – 6,552.
  2. 2

    Divide your number by the smallest possible prime factor. Divide your number by the smallest prime factor (besides 1) that divides into it evenly with no remainder.

    Write the prime factor in the left column and write your answer across from it in the right column. As noted above, even numbers are especially easy to start factoring because their smallest prime factor will always be 2.

    Odd numbers, on the other hand, will have smallest prime factors that differ.

    • In our example, since 6,552 is even, we know that 2 is its smallest prime factor. 6,552 ÷ 2 = 3,276. In the left column, we'll write 2, and in the right column, write 3,276.
  3. 3

    Continue to factor in this fashion. Next, factor the number in the right column by its smallest prime factor, rather than the number at the top of the table. Write the prime factor in the left column and the new number in the right column. Continue to repeat this process – with each repetition, the number in the right column should decrease.

    • Let's continue with our process. 3,276 ÷ 2 = 1,638, so att the bottom of the left column, we'll write another 2, and at the bottom of the right column, we'll write 1,638. 1,638 ÷ 2 = 819, so we'll write 2 and 819 at the bottom of the two columns as before.
  4. 4

    Deal with odd numbers by trying small prime factors.

    Odd numbers are more difficult to find the smallest prime factor of than even numbers because they don't automatically have 2 as their smallest prime factor.

    When you reach an odd number, try dividing by small prime numbers other than 2 – 3, 5, 7, 11, and so on – until you find one that divides evenly with no remainder. This is the number's smallest prime factor.[5]

    • In our example, we've reached 819. 819 is odd, so 2 is not a factor of 819. Instead of writing down another 2, we'll try the next prime number: 3. 819 ÷ 3 = 273 with no remainder, so we'll write down 3 and 273.
    • When guessing factors, you should try all prime numbers up to the square root of the largest factor found so far. If none of the factors you try up to this point divide evenly, you're probably trying to factor a prime number and thus are finished with the factoring process.
  5. 5

    Continue until you reach 1. Continue dividing the numbers in the right column by their smallest prime factor until you obtain a prime number in the right column. Divide this number by itself – this will put the number in the left column and “1” in the right column.

    • Let's finish factoring our number. See below for a detailed breakdown:
      • Divide by 3 again: 273 ÷ 3 = 91, no remainder, so we'll write down 3 and 91.
      • Let's try 3 again: 91 doesn't have 3 as a factor, nor does it have the next lowest prime (5) as a factor, but 91 ÷ 7 = 13, with no remainder, so we'll write down 7 and 13.
      • Let's try 7 again: 13 doesn't have 7 as a factor, or 11 (the next prime), but it does have itself as a factor: 13 ÷ 13 = 1. So, to finish our table, we'll write down 13 and 1. We can finally stop factoring.
  6. 6

    Use the numbers in the left-hand column as your original number's factors. Once you reach 1 in the right-hand column, you're done. The numbers listed on the left side of the table are your factors.

    In other words, the product when you multiply all of these numbers together will be the number at the top of the table. If the same factor appears multiple times, you can use exponent notation to save space.

    For instance, if your list of factors has four 2's, you can write 24 rather than 2 × 2 × 2 × 2.

    • In our example 6,552 = 23 × 32 × 7 × 13. This is the complete factorization of 6,552 into prime numbers. No matter what order these numbers are multiplied in, the product will be 6,552.

All Factors of a Number

Go straight to Factors Calculator.

Factors are the numbers you multiply together
to get another number:

There can be many factors of a number.

  • 2 × 6 = 12,
  • but also 3 × 4 = 12,
  • and of course 1 × 12 = 12.
  • So 1, 2, 3, 4, 6 and 12 are factors of 12.
  • And also -1,-2,-3,-4,-6 and -12, because you get a positive number when you multiply two negatives, such as (-2)×(-6) = 12
  • Answer: 1, 2, 3, 4, 6, 12, -1, -2, -3, -4, -6, -12

No Fractions!

Factors are usually positive or negative whole numbers (no fractions), so ½ × 24 = 12 is not listed.

All Factors Calculator

This calculator will find all the factors of a number (not just the prime factors). It works on numbers up to 4,294,967,295. Try it and see.

Note: Negative numbers are also included, as multiplying two negatives makes a positive.

How Can I Do It Myself?

Work from the outside in!

Example: All the factors of 20

Start at 1: 1×20=20, so put 1 at the start, and put its “partner” 20 at the other end:

Then try 2. 2×10=20 works, so put in 2 and 10:

Then try 3. 3 doesn't work (3×6=18 is too low, 3×7=21 is too high).

Then try 4. 4×5=20 works, so put them in:

There is no whole number between 4 and 5 so you are done! (Don't forget the negative ones).

1 2 4 5 10 20
-1 -2 -4 -5 -10 -20

Is That How The Calculator Works?

Actually the calculator first works out the prime factors, then combines them together to discover all other numbers that can be multiplied to achieve your number.

Copyright © 2019 MathsIsFun.com

How to Find All The Factors of a Number Quickly and Easily

Updated May 10, 2018

By Claire Gillespie

Finding the factors of a number is an important math skill for basic arithmetic, algebra and calculus. The factors of a number are any numbers that divide into it exactly, including 1 and the number itself. In other words, every number is the product of multiple factors.

The quickest way to find the factors of a number is to divide it by the smallest prime number (bigger than 1) that goes into it evenly with no remainder. Continue this process with each number you get, until you reach 1.

A number that can only be divided by 1 and itself is called a prime number. Examples of prime numbers are 2, 3, 5, 7, 11 and 13. The number 1 is not considered a prime number because 1 goes into everything.

Some divisibility rules can help you find the factors of a number. If a number is even, it's divisible by 2, i.e. 2 is a factor. If a number's digits total a number that's divisible by 3, the number itself is divisible by 3, i.e. 3 is a factor. If a number ends with a 0 or a 5, it's divisible by 5, i.e. 5 is a factor.

If a number is divisible twice by 2, it's divisible by 4, i.e. 4 is a factor. If a number is divisible by 2 and by 3, it's divisible by 6, i.e. 6 is a factor. If a number is divisible twice by 3 (or if the sum of the digits is divisible by 9), then it's divisible by 9, i.e. 9 is a factor.

Establish the number you want to find the factors of, for example 24. Find two more numbers that multiply to make 24. In this case, 1 x 24 = 2 x 12 = 3 x 8 = 4 x 6 = 24. This means the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

Factor negative numbers in the same way as positive numbers, but make sure the factors multiply together to produce a negative number. For example, the factors of -30 are -1, 1, -2, 2, -3, 3, -5, 5, -6, 6, -10, 10, -15 and 15.

If you have a large number, it's more difficult to do the mental math to find its factors. To make it easier, create a table with two columns and write the number above it.

Using the number 3784 as an example, start by dividing it by the smallest prime factor (bigger than 1) that goes into it evenly with no remainder. In this case, 2 x 1892 = 3784.

Write the prime factor (2) in the left column and the other number (1892) in the right column.

Continue with this process, i.e. 2 x 946 = 1892, adding both numbers to the table. When you reach an odd number (e.g., 2 x 473 = 946), divide by small prime numbers besides 2 until you find one that divides evenly with no remainder. In this case, 11 x 43 = 473. Continue the process until you reach 1.

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Factoring Calculator

The Factoring Calculator finds the factors and factor pairs of a positive or negative number. Enter an integer number to find its factors.

For positive integers the calculator will only present the positive factors because that is the normally accepted answer. For example, you get 2 and 3 as a factor pair of 6.

If you also need negative factors you will need to duplicate the answer yourself and repeat all of the factors as negatives such as -2 and -3 as another factor pair of 6.

On the other hand this calculator will give you negative factors for negative integers. For example, -2 and 3 AND 2 and -3 are both factor pairs of -6.

Factors are whole numbers that are multiplied together to produce another number. The original numbers are factors of the product number. If a x b = c then a and b are factors of c.

Say you wanted to find the factors of 16. You would find all pairs of numbers that when multiplied together resulted in 16. We know 2 and 8 are factors of 16 because 2 x 8 = 16. 4 is a factor of 16 because 4 x 4 = 16. Also 1 and 16 are factors of 16 because 1 x 16 = 16. The factors of 16 are 1, 2, 4, 8, 16.

You can also think about factors in terms of division: The factors of a number include all numbers that divide evenly into that number with no remainder. Consider the number 10. Since 10 is evenly divisible by 2 and 5, you can conclude that both 2 and 5 are factors of 10.

The table below lists the factors for 3, 18, 36 and 48. It is important to note that every integer number has at least two factors: 1 and the number itself. If a number has only two factors that number is a prime number.

1, 2, 3, 4, 6, 9, 12, 18, 36 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

How to Factor Numbers: Factorization

This factors calculator factors numbers by trial division. Follow these steps to use trial division to find the factors of a number.

  1. Find the square root of the integer number n and round down to the closest whole number. Let's call this number s.
  2. Start with the number 1 and find the corresponding factor pair: n ÷ 1 = n. So 1 and n are a factor pair because division results in a whole number with zero remainder.
  3. Do the same with the number 2 and proceed testing all integers (n ÷ 2, n ÷ 3, n ÷ 4… n ÷ s) up through the square root rounded to s. Record the factor pairs where division results in whole integer numbers with zero remainders.
  4. When you reach n ÷ s and you have recorded all factor pairs you have successfully factored the number n.

Example Factorization Using Trial Division

Factors of 18:

  • The square root of 18 is 4.2426, rounded down to the closest whole number is 4
  • Testing the integer values 1 through 4 for division into 18 with a 0 remainder we get these factor pairs: (1 and 18), (2 and 9), (3 and 6). The factors of 18 are 1, 2, 3, 6, 9, 18.

Factors of Negative Numbers

All of the above information and methods generally apply to factoring negative numbers. Just be sure to follow the rules of multiplying and dividing negative numbers to find all factors of negative numbers. For example, the factors of -6 are (1, -6), (-1, 6), (2, -3), (-2, 3). See the Math Equation Solver Calculator and the section on Rules for Multiplication Operations.

Further Reading

Are Negatives Factors of Other Numbers? at http://mathforum.org/library/drmath/view/63516.html

Related Factoring Calculators

  • See our Common Factors Calculator to find all factors of a set of numbers and learn which are the common factors.
  • The Greatest Common Factor Calculator finds the greatest common factor (GCF) or greatest common divisor (GCD) of a set of numbers.
  • See the Least Common Denominator Calculator to find the lowest common denominator for fractions, integers and mixed numbers.

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