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Factors of 420: Prime Factorization, Methods, and Examples

The factors of 420 are those numbers that give a zero remainder when divided by zero. There are 24 factors of 420. 

This article finds out all the factors of 420 and details how they are calculated.

Factors of 420

Here are the factors of number 420.

Factors of 420: 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20 , 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210 and 420.

Negative Factors of 420

The negative factors of 420 are similar to its positive aspects, just with a negative sign.

Negative Factors of 420: 1, -2, -3, -4, -5, -6, -7, -10, -12, -14, -15, -20 , -21, -28, -30, -35, -42, -60, -70, -84, -105, -140, -210 and -420.

Prime Factorization of 420

The prime factorization of 420 is the way of expressing its prime factors in the product form.

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

In this article, we will learn about the factors of 420 and how to find them using various techniques such as upside-down division, prime factorization, and factor tree.

What Are the Factors of 420?

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

The factors of 420 are classified as prime numbers and composite numbers. The prime factors of the number 420 can be determined using the prime factorization technique.

How To Find the Factors of 420?

You can find the factors of 420 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 420, create a list containing the numbers that are exactly divisible by 420 with zero remainders. One important thing to note is that 1 and 420 are the 420’s factors, as every natural number has one and the number itself as its factor.

One is also called the universal factor of every number. The factors of 420 are determined as follows:

\[\dfrac{420}{1} = 420\]

\[\dfrac{420}{2} = 210\]

\[\dfrac{420}{3} = 140\]

\[\dfrac{420}{4} = 105\]

\[\dfrac{420}{5} = 84\]

\[\dfrac{420}{6} = 70\]

\[\dfrac{420}{7} = 60\]

\[\dfrac{420}{10} = 42\]

\[\dfrac{420}{12} = 35\]

\[\dfrac{420}{14} = 30\]

\[\dfrac{420}{15} = 28\]

\[\dfrac{420}{20} = 21\]

\[\dfrac{420}{420} = 1\]

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

Total Number of Factors of 420

For 420, there are 24 positive factors and 24 negative ones. So in total, there are 48 factors of 420. 

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 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 420 is given as:

The factorization of 420 is 1 x 22 × 3 × 5 × 7.

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

Adding 1 to each and multiplying them together results in 48.

Therefore, the total number of factors of 420 is 48. 24 are positive, and 24 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 number factors 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 two as its prime factor, the smallest prime factor.

Factors of 420 by Prime Factorization

The number 420 is 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 420 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 420, divide by its most minor prime factor. First, determine that the given number is either even or odd. Two will be the smallest prime factor if it is an even number.

Continue splitting the quotient obtained until 1 is received as the quotient. The prime factorization of 420 can be expressed as:

420 = 22 × 3 × 5 × 7

Factors of 420 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.

For 420, the factor pairs can be found as:

1 x 420 = 420

2 x 210 = 420

3 x 140 = 420

4 x 105 = 420

5 x 84 = 420

6 x 70 = 420

7 x 60 = 420

10 x 42 = 420

12 x 35 = 420

14 x 30 = 420

15 x 28 = 420

20 x 21 = 420

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

All these numbers in pairs, when multiplied, give 420 as the product.

The negative factor pairs of 420 are given as:

-1 x -420 = 420

-2 x -210 = 420

-3 x -140 = 420

-4 x -105 = 420

-5 x -84 = 420

-6 x -70 = 420

-7 x -60 = 420

-10 x -42 = 420

-12 x -35 = 420

-14 x -30 = 420

-15 x -28 = 420

-20 x -21 = 420

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, -10, -12, -14, -15, -20 , -21, -28, -30, -35, -42, -60, -70, -84, -105, -140, -210 and -420 are called negative factors of 420.

The list of all the factors of 420, including positive and negative numbers, is given below.

Factor list of 420: 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 10, -10, 12, -12, 14, -14, 15, -15, 20, -20 , 21, -21, 28, -28, 30, -30, 35, -35, 42, -42, 60, -60, 70, -70, 84, -84, 105, -105, 140, -140, 210, -210, 420, and -420.

Factors of 420 Solved Examples

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

Example 1

How many factors of 420 are there?

Solution

The total number of Factors of 420 is 48.

Factors of 420 are 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20 , 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210 and 420.

Example 2

Find the factors of 420 using prime factorization.

Solution

The prime factorization of 420 is given as:

420 $\div$ 7 = 60 

60 $\div$ 5 = 12 

12 $\div$ 3 = 4 

4 $\div$ 4 = 1 

So the prime factorization of 420 can be written as:

22 × 3 × 5 × 7 = 420

Factors of 419|Factors List| Factors of 421

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