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Factors of 312: Prime Factorization, Methods, and Examples

The factors of 312 are numbers that, when divided by 312, leave zero as the remainder. This means the numbers that ultimately divide the given number are named as their factors. The given number’s factors can be positive and negative, provided that the given number is achieved upon multiplication of two-factor integers.

Factors of 312

Here are the factors of number 312.

Factors of 312: 1, 2, 3, 4, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, 156, and 312

Negative Factors of 312

The negative factors of 312 are similar to its positive aspects, just with a negative sign.

Negative Factors of 312: –1, -2, -3, -4, -6, -8, -12, -13, -24, -26, -39, -52, -78, -104, -156, and -312

Prime Factorization of 312

The prime factorization of 312 is the way of expressing its prime factors in the product form.

Prime Factorization: 2 x 2 x 2 x 3 x 13 

In this article, we will learn about the factors of 312 and how to find them using various techniques such as upside-down division, prime factorization, and factor tree.

What Are the Factors of 312?

The factors of 312 are 1, 2, 3, 4, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, 156, and 312. These numbers are the factors as they do not leave any remainder when divided by 312.

The factors of 312 are classified as prime numbers and composite numbers. The prime factors of the number 312 can be determined using the prime factorization technique.

How To Find the Factors of 312?

You can find the factors of 312 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 312, create a list containing the numbers that are exactly divisible by 312 with zero remainders. One important thing to note is that 1 and 312 are 312’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 312 are determined as follows:

\[\dfrac{312}{1} = 312\]

\[\dfrac{312}{2} = 156\]

\[\dfrac{312}{3} = 104\]

\[\dfrac{312}{4} = 78\]

\[\dfrac{312}{6} = 52\]

\[\dfrac{312}{8} = 39\]

\[\dfrac{312}{12} = 26\]

\[\dfrac{312}{13} = 24\]

\[\dfrac{312}{312} = 1\]

Therefore, 1, 2, 3, 4, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, 156, and 312 are the factors of 312.

Total Number of Factors of 312

For 312, there are 16 positive factors and 16 negative ones. So in total, there are 32 factors of 312. 

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 312 is given as:

Factorization of 312 is 1 x 2$^3$ x 3 x 13.

The exponent of 1, 3, and 13 is 1. The exponent of 2 is 3.

Adding 1 to each and multiplying them together results in 32.

Therefore, the total number of factors of 312 is 32. Sixteen are positive, and sixteen 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 312 by Prime Factorization

The number 312 is a composite. 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 312 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 312, 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 312 can be expressed as:

312 = 2 x 2 x 2 x 3 x 13

Factors of 312 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 312, the factor pairs can be found as:

1 x 312 = 312

2 x 156 = 312 

3 x 104 = 312 

4 x 78 = 312 

6 x 52 = 312 

8 x 39 = 312 

12 x 26 = 312 

13 x 24 = 312 

The possible factor pairs of 312 are given as (1, 312), (2, 156), (3, 104), (4, 78), (6, 52), (8, 39), (12, 26), and (13, 24).

All these numbers in pairs, when multiplied, give 312 as the product.

The negative factor pairs of 312 are given as:

-1 x -312 = 312

-2 x -156 = 312 

-3 x -104 = 312 

-4 x -78 = 312 

-6 x -52 = 312 

-8 x -39 = 312 

-12 x -26 = 312 

-13 x -24 = 312 

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, -6, -8, -12, -13, -24, -26, -39, -52, -78, -104, -156, and -312 are called negative factors of 312.

Factors of 312 Solved Examples

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

Example 1

How many factors of 312 are there?

Solution

The total number of Factors of 312 is sixteen.

Factors of 312 are 1, 2, 3, 4, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, 156, and 312.

Example 2

Find the factors of 312 using prime factorization.

Solution

The prime factorization of 312 is given as:

312 $\div$ 2 = 156 

156 $\div$ 2 = 78 

78 $\div$ 2 = 39

39 $\div$ 3 = 13

13 $\div$ 13 = 1 

So the prime factorization of 312 can be written as:

2 x 2 x 2 x 3 x 13 = 312

Factors of 311|Factors List| Factors of 313

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