Factors of 408: Prime Factorization, Methods, and Examples

The factors of 408 are numbers that divide 408 evenly with zero remainders. The number 408 is an even composite with a total of 16 factors. It has three prime factors. 

Let us explore more about its factors. 

Factors of 408

Here are the factors of number 408.

Factors of 408: 1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, 136, 204, and 408  

Negative Factors of 408

The negative factors of 408 are similar to its positive aspects, just with a negative sign.

Negative Factors of 408: –1, -2, -3, -4, -6, -8, -12, -17, -24, -34, -51, -68, -102, -136, -204, and -408  

Prime Factorization of 408

The prime factorization of 408 is the way of expressing its prime factors in the product form.

Prime Factorization: 2 x 2 x 2 x 3 x 17

In this article, we will learn about the factors of 408 and how to find them using various techniques such as upside-down division, prime factorization, and factor tree.

What Are the Factors of 408?

The factors of 408 are 1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, 136, 204, and 408. These numbers are the factors as they do not leave any remainder when divided by 408.

The factors of 408 are classified as prime numbers and composite numbers. The prime factors of the number 408 can be determined using the prime factorization technique.

How To Find the Factors of 408?

You can find the factors of 408 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 408, create a list containing the numbers that are exactly divisible by 408 with zero remainders. One important thing to note is that 1 and 408 are 408’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 408 are determined as follows:

\[\dfrac{408}{1} = 408\]

\[\dfrac{408}{2} = 204\]

\[\dfrac{408}{3} = 136\]

\[\dfrac{408}{4} = 102\]

\[\dfrac{408}{6} = 68\]

\[\dfrac{408}{8} = 51\]

\[\dfrac{408}{12} = 34\]

\[\dfrac{408}{17} = 24\]

Therefore, 1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, 136, 204, and 408 are the factors of 408.

Total Number of Factors of 408

For 408, there are sixteen positive factors and sixteen negative ones. So in total, there are thirty-two factors of 408. 

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 408 is given as:

Factorization of 408 is 1 x 2$^3$ x 3 x 17.

The exponent of 1, 3, and 17 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 408 is 32. Sixteen of them 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 408 by Prime Factorization

The number 408 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 408 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 408, 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 408 can be expressed as:

408 = 2 x 2 x 2 x 3 x 17

Factors of 408 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 408, the factor pairs can be found as:

1 x 408 = 408

2 x 204 = 408 

3 x 136 = 408

4 x 102 = 408  

6 x 68 = 408 

8 x 51 = 408

12 x 34 = 408

17 x 24 = 408   

The possible factor pairs of 408 are given as (1, 408), (2, 204), (3, 136), (4, 102), (6, 68), (8, 51), (12, 34), and (17, 24).

All these numbers in pairs, when multiplied, give 408 as the product.

The negative factor pairs of 408 are given as:

-1 x -408 = 408

-2 x -204 = 408 

-3 x -136 = 408

-4 x -102 = 408  

-6 x -68 = 408 

-8 x -51 = 408

-12 x -34 = 408

-17 x -24 = 408   

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, -17, -24, -34, -51, -68, -102, -136, -204, and -408 are called negative factors of 408.

The list of all the factors of 408, including positive as well as negative numbers, is given below.

Factor list of 408: 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 8, -8, 12, -12, 17, -17, 24, -24, 34, -34, 51, -51, 68, -68, 102, -102, 136, -136, 204, -204, 408 and -408.

Factors of 408 Solved Examples

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

Example 1

How many factors of 408 are there?

Solution

The total number of Factors of 408 is 16.

Factors of 408 are 1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, 136, 204, and 408.

Example 2

Find the factors of 408 using prime factorization.

Solution

The prime factorization of 408 is given as:

408 $\div$ 2 = 204 

204 $\div$ 2 = 102

102 $\div$ 2 = 51

51 $\div$ 3 = 17   

17 $\div$ 17 = 1 

So the prime factorization of 408 can be written as:

2 x 2 x 2 x 3 x 17 = 408

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