# Factors of 840: Prime Factorization, Methods, and Examples

The factors of 840 are a set of numbers that when divided by 840 leave a remainder of zero. So the numbers that ultimately divide 840 are its factors.

840 is an even composite number and has a total of 32 factors.

### Factors of 840

Here are the factors of number 840.

Factors of 840: 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840

### Negative Factors of 840

The negative factors of 840 are similar to their positive aspects, just with a negative sign.

Negative Factors of 840: -1, -2, -3, -4, -5, -6, -7, -8, -10, -12, -14, -15, -20, -21, -24, -28, -30, -35, -40, -42, -56, -60, -70, -84, -105, -120, -140, -168, -210, -280, -420, -840

### Prime Factorization of 840

The prime factorization of 840 is the way of expressing its prime factors in the product form.

Prime Factorization: 2 x 2 x 2 x 3 x 5 x 7

In this article, we will learn about the factors of 840 and how to find them using various techniques such as upside-down division, prime factorization, and factor tree.

## What Are the Factors of 840?

The factors of 840 are 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, and 840. These numbers are the factors as they do not leave any remainder when divided by 840.

The factors of 840 are classified as prime numbers and composite numbers. The prime factors of the number 840 can be determined using the prime factorization technique.

## How To Find the Factors of 840?

You can find the factors of 840 by using the rules of divisibility. The divisibility rule states that any number, when divided by any other natural number, is said to be divisible by the number if the quotient is the whole number and the resulting remainder is zero.

To find the factors of 840, create a list containing the numbers that are exactly divisible by 840 with zero remainders. One important thing to note is that 1 and 840 are the 840’s factors as every natural number has 1 and the number itself as its factor.

1 is also called the universal factor of every number. The factors of 840 are determined as follows:

$\dfrac{840}{1} = 840$

$\dfrac{840}{2} = 420$

$\dfrac{840}{3} = 280$

$\dfrac{840}{4} = 210$

$\dfrac{840}{5} = 168$

$\dfrac{840}{6} = 140$

$\dfrac{840}{7} = 120$

$\dfrac{840}{8} = 105$

$\dfrac{840}{10} = 84$

$\dfrac{840}{12} = 70$

$\dfrac{840}{14} = 60$

$\dfrac{840}{15} = 56$

$\dfrac{840}{20} = 42$

$\dfrac{840}{21} = 40$

$\dfrac{840}{24} = 35$

$\dfrac{840}{28} = 30$

$\dfrac{840}{840} = 1$

Therefore,1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, and 840 are the factors of 840.

### Total Number of Factors of 840

For 840, there are 32 positive factors and 32 negative ones. So in total, there are 64 factors of 840.

To find the total number of factors of the given number, follow the procedure mentioned below:

1. Find the factorization/prime factorization of the given number.
2. Demonstrate the prime factorization of the number in the form of exponent form.
3. Add 1 to each of the exponents of the prime factor.
4. Now, multiply the resulting exponents together. This obtained product is equivalent to the total number of factors of the given number.

By following this procedure, the total number of factors of 840 is given as:

Factorization of 840 is 1 x 2 x 2 x 2 x 3 x 5 x 7.

The exponent of 1, 2, 3, 5, and 7 is 1, 3, 1, 1, and 1 respectively.

Adding 1 to each and multiplying them together results in 64.

Therefore, the total number of factors of 840 is 64. 32 is positive, and 32 factors are negative.

### Important Notes

Here are some essential points that must be considered while finding the factors of any given number:

• The factor of any given number must be a whole number.
• The factors of the number cannot be in the form of decimals or fractions.
• Factors can be positive as well as negative.
• Negative factors are the additive inverse of the positive factors of a given number.
• The factor of a number cannot be greater than that number.
• Every even number has 2 as its prime factor, the smallest prime factor.

## Factors of 840 by Prime Factorization

The number 840 is a composite number. Prime factorization is a valuable technique for finding the number’s prime factors and expressing the number as the product of its prime factors.

Before finding the factors of 840 using prime factorization, let us find out what prime factors are. Prime factors are the factors of any given number that are only divisible by 1 and themselves.

To start the prime factorization of 840, start dividing by its most minor prime factor. First, determine that the given number is either even or odd. If it is an even number, then 2 will be the smallest prime factor.

Continue splitting the quotient obtained until 1 is received as the quotient. The prime factorization of 840 can be expressed as:

840 = 2 x 2 x 2 x 3 x 5 x 7

## Factors of 840 in Pairs

The factor pairs are the duplet of numbers that, when multiplied together, result in the factorized number. Factor pairs can be more than one depending on the total number of factors given.

For 840, the factor pairs can be found as:

1 x 840 = 840

2 x 420 = 840

3 x 280 = 840

4 x 210 = 840

5 x 168 = 840

6 x 140 = 840

7 x 120= 840

8 x 105 = 840

10 x 84 = 840

12 x 70 = 840

14 x 60 = 840

15 x 56 = 840

20 x 42 = 840

21 x 40 = 840

24 x 35 = 840

28 x 30 = 840

The possible factor pairs of 840 are given as (1, 840), (2, 420), (3, 280), (4, 210), (5, 168), (6, 140), (7, 120), (8, 105), (10, 84), (12, 70), (14, 60), (15, 56), (20, 42), (21, 40), (24, 35) and (28, 30).

All these numbers in pairs, when multiplied, give 840 as the product.

The negative factor pairs of 840 are given as:

-1 x -840 = 840

-2 x -420 = 840

-3 x -280 = 840

-4 x -210 = 840

-5 x -168 = 840

-6 x -140 = 840

-7 x -120= 840

-8 x -105 = 840

-10 x -84 = 840

-12 x -70 = 840

-14 x -60 = 840

-15 x -56 = 840

-20 x -42 = 840

-21 x -40 = 840

-24 x -35 = 840

-28 x -30 = 840

It is important to note that in negative factor pairs, the minus sign has been multiplied by the minus sign, due to which the resulting product is the original positive number. Therefore, -1, -2, -3, -4, -5, -6, -7, -8, -10, -12, -14, -15, -20, -21, -24, -28, -30, -35, -40, -42, -56, -60, -70, -84, -105, -120, -140, -168, -210, -280, -420, -840 are called negative factors of 840.

The list of all the factors of 840, including positive as well as negative numbers, is given below.

Factor list of 840: 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8, -8, 10, -10, 12, -12, 14,- 14, 15, -15, 20, -20, 21, -21, 24, -24, 28, -28, 30, -30, 35, -35, 40, -40, 42, -42, 56, -56, 60, -60, 70, -70, 84, -84, 105, -105, 120, -120, 140, -140, 168, -168, 210, -210, 280, -280, 420, -420, 840, and -840

## Factors of 840 Solved Examples

To better understand the concept of factors, let’s solve some examples.

### Example 1

How many factors of 840 are there?

### Solution

The total number of Factors of 840 is 64.

Factors of 840 are 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, and 840.

### Example 2

Find the factors of 840 using prime factorization.

### Solution

The prime factorization of 840 is given as:

840 $\div$ 2 = 420

420 $\div$ 2 = 210

210 $\div$ 2 = 105

105 $\div$ 3 = 35

35 $\div$ 5 = 7

7 $\div$ 7 = 1

So the prime factorization of 840 can be written as:

2 x 2 x 2 x 3 x 5 x 7 = 840