Factoring Calculator

Factoring calculator

A factoring calculator is an online tool allowing you to quickly find all factors of any integer (except for 0). Since integers are whole numbers that can be positive or negative, we can use this factor finder for both positive and negative numbers.

Limitations on the input values of the factor calculator:

• You can only input integers (positive or negative).
• Entering 0 is not possible.

Directions for use

To find all factors of a number, enter that number and press “Calculate.” The factors calculator will return the list of factors of the number and the total number of factors. The calculator will also return the factor pairs of the number.

Factorization: definitions and formulas

In mathematics, factorization is defined as a process of dividing an object into a multiplication of several other objects or factors. Various mathematical objects, such as numbers, polynomials, and matrices, can be factorized. Here we will focus on the factorization of integers.

The factors of an integer are such integers that they divide the given integer without a remainder.

Basically, for non-zero integers a, b and c, if a = b × c, then b and c are the factors of a. For example, 1, 2, 3 and 6 are all factors of 6, since they all divide 6 evenly (without any remainder):

• 6 / 1 = 6
• 6 / 2 = 3
• 6 / 3 = 2
• 6 / 6 = 1

Any integer will always have at least two factors: 1 and the integer itself, i.e., any a can be factorized as a = 1 × a.

How to find factors of a number

The calculator uses the trial division method to find the factors of any given number. This is the most straightforward integer factorization algorithm, which consistently tests whether the number is evenly divided by all numbers smaller than the given number itself.

There are several ways to make the process less cumbersome. First, the numbers are always tested in increasing order, starting with 2. Then, suppose 2 is not the factor of the given number. In that case, the multiples of 2 are automatically discarded, and the process becomes easier.

Furthermore, for the given a, you should only perform up the testing until √a. This is true since, if b is a factor of a, such that a = b × c. Then, if c were smaller than b, it would have already been identified as the factor of a.

We can reduce the mechanism to the following steps:

For the given number a, find the square root of a: √a, and round it down to the closest whole number. Let’s denote the rounded-down square root of a as r.

Test out all integers greater than or equal to 1 and less than or equal to r to see if they are evenly dividing a. Remember that if you have already established that a prime number is not one of the factors of the given number, you don’t have to check the multiples of this prime number anymore! For example, if you have found that the given number cannot be equally divided by 3, you can skip all multiples of 3, such as 6, 9 and so on.

Write down all factors and the corresponding factor pairs.

Calculation example

The parents are planning a birthday party for their son, Mike, who is turning 6. At the end of the party, they want to give sweet treats to every child attending. They have prepared 32 cupcakes to give to the children.

How many guests can Mike invite to his party, so each guest receives the same number of treats at the end of the celebration? How many cupcakes will each child get?

Solution

We must find how many guests Mike can invite to the party so that each guest receives the same number of cupcakes from the 32 available ones. We have to find which whole numbers divide 32 without remainders (so that the cupcakes don’t have to be broken into pieces). This means we have to find all positive factors of 32. To determine how many cupcakes each child will get in every case, we also have to find the factor pairs.

Let’s use the trial division method to find the factors and the factor pairs of the given number. As the first step, we have to find the square root of the number:

$$\sqrt{32}\approx5.657$$

Rounding 5.657 down to the next integer, we get 5. This means we have to check all whole numbers greater than or equal to 1 and less than or equal to 5.

For number 1:

32 / 1 = 32. 1 is a factor of 32, since 1 is a factor of any integer: 1 × 32 = 32. So, if Mike only has one guest, they will get all 32 cupcakes! Alternatively, if he decides to invite 32 children to his party, each child will only get one cupcake in the evening.

For number 2:

32 / 2 = 16. This means that 2 is a factor of 32. The corresponding factor pair is: 2 × 16 = 32. Also, here, both 2 and 16 are the factors of 32 and have to be included in the list of factors, meaning that if Mike invites two guests, they will get 16 cupcakes each. But if he invites 16 children, each of them will receive 2 cupcakes at the end of the party.

For number 3:

32 / 3 = 10 2/3 ≅ 10.667. This means that 3 does not evenly divide 32 and is not a factor of 32. Mike cannot invite 3 guests to his party since in that case, the division of cupcakes would be unfair.

Since 2 was a factor of the given number, we cannot skip the multiples of 2, and we have to check 4 as well.

For number 4:

32 / 4 = 8. This means that 4 is a factor of 32. The corresponding factor pair is: 4 × 8 = 32. Mike can invite 4 children, in which case each child will get 8 cupcakes, or he can invite 8 children, then each guest will get 4 cupcakes.

For number 5:

32 / 5 = 6 2/5 = 6.4. This means that 5 does not evenly divide 32 and is not a factor of 32. So, inviting 5 guests is also not an option for Mike.

Since we only had to check integers greater than or equal to 1 and less than or equal to 5, we have found all the given number factors!

Answer

The six factors of 32 are:

1, 2, 4, 8, 16, 32

Mike can invite 1, 2, 4, 8, 16, or 32 guests to his party for the cupcake distribution to be fair.

The factor pairs of 32 are:

• 1 × 32 = 32

• 2 × 16 = 32

• 4 × 8 = 32

In each factor pair, one of the numbers represents the number of guests, and the other number represents the number of cupcakes each guest will receive at the end of the party.