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Factors of 784: Prime Factorization, Methods, and Examples

The numbers that make up 784’s factors result in a whole number quotient and a remainder of zero when multiplied by 784. It indicates that 784’s components are all the integers that divide it in half. 

The next paragraphs provide a detailed explanation of the elements, which can be both good and negative.

Factors of 784

Here are the factors of number 784.

Factors of 784: 1, 2, 4, 7, 8, 14, 16, 28, 49, 56, 98, 112, 196, 392 and 784

Negative Factors of 784

The negative factors of 784 are similar to their positive aspects, just with a negative sign.

Negative Factors of 784: -1,- 2, -4,- 7, -8, -14, -16, -28, -49, -56, -98, -112, -196, -392 and -784

Prime Factorization of 784

The prime factorization of 784 is the way of expressing its prime factors in the product form.

Prime Factorization: 24 x 72

In this article, we will learn about the factors of 784and how to find them using various techniques such as upside-down division, prime factorization, and factor tree.

What Are the Factors of 784?

The factors of 784 are 1, 2, 4, 7, 8, 14, 16, 28, 49, 56, 98, 112, 196, 392 and 784. These numbers are the factors as they do not leave any remainder when divided by 784.

The factors of 784 are classified as prime numbers and composite numbers. The prime factors of the number 784 can be determined using the prime factorization technique.

How To Find the Factors of 784?

You can find the factors of 784 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 784, create a list containing the numbers that are exactly divisible by 784 with zero remainders. One important thing to note is that 1 and 784 are the 784’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 784 are determined as follows:

\[\dfrac{ 784}{1} = 784\]

\[\dfrac{ 784}{2} = d\]

\[\dfrac{ 784}{4} = e\]

\[\dfrac{ 784}{ 7} = 1\]

\[\dfrac{ 784}{8} = 784\]

\[\dfrac{ 784}{14} = d\]

\[\dfrac{ 784}{16} = e\]

\[\dfrac{ 784}{ 28} = 1\]

\[\dfrac{ 784}{49} = 784\]

\[\dfrac{ 784}{56} = d\]

\[\dfrac{ 784}{98} = e\]

\[\dfrac{ 784}{112} = e\]

[\dfrac{ 784}{196} = e\]

\[\dfrac{ 784}{392} = e\]

\[\dfrac{ 784}{ 784} = 1\]

Therefore, 1, 2, 4, 7, 8, 14, 16, 28, 49, 56, 98, 112, 196, 392 and 784 are the factors of 784.

Total Number of Factors of 784

For 784, there are 15 positive factors and 15 negative ones. So in total, there are 30 factors of 784. 

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 784 is given as:

Factorization of 784 is 1 x 24 x 72.

The exponent of 1 is 1, 2 is 4, and 7 is 2.

Adding 1 to each and multiplying them together results in 30.

Therefore, the total number of factors of 784 is 30. 15 are positive, and 15 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 784 by Prime Factorization

The number 784 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 784 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 784, 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 784 can be expressed as:

 784 = 24 x 72

Factors of 784 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 784, the factor pairs can be found as:

1 x 784 = 784

2 x 392 = 784

4 x 196 = 784

7 x 112 = 784

8 x 98 = 784

14 x 56 = 784

16 x 49 = 784

28 x 28 = 784

The possible factor pairs of 784 are given as (1, 784),(2, 392),(4, 196),(7, 112),(8, 98),(14, 56),(16, 49), and (28, 28).

All these numbers in pairs, when multiplied, give 784 as the product.

The negative factor pairs of 784 are given as:

-1 x – 784 = 784

-1 x -784 = 784

-2 x -392 = 784

-4 x -196 = 784

-7 x -112 = 784

-8 x -98 = 784

-14 x -56 = 784

-16 x -49 = 784

-28 x -28 = 784

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, -4,- 7, -8, -14, -16, -28, -49, -56, -98, -112, -196, -392 and -784 are called negative factors of 784.

The list of all the factors of 784, including positive as well as negative numbers, is given below.

Factor list of 784: 1,-1,2,-2, 4,-4,7,-7, 8,-8,14, -14,16, -16,28, -28,49, -49, 56,-56,98, -98,112, -112,196, -196,392, -392 , 784, and – 784

Factors of 784 Solved Examples

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

Example 1

How many factors of 784 are there?

Solution

The total number of Factors of 784 is 30.

Factors of 784 are 1, 2, 4, 7, 8, 14, 16, 28, 49, 56, 98, 112, 196, 392 and 784.

Example 2

Find the factors of 784 using prime factorization.

Solution

The prime factorization of 784 is given as:

784 $\div$ 2 = 392

392 $\div$ 2 = 196 

196 $\div$ 2 = 98 

98 $\div$ 2 = 49 

49 $\div$ 7 = 7 

So the prime factorization of 784 can be written as:

24 x 72 = 784

Factors of 783|Factors List| Factors of 785

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