Factors of 420

The numbers that divide 420 evenly are known as factors of 420.

A factor of 420 is a number that divides the number without a remainder.

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.

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 factors of 420: 2, 3, 5, and 7.

Prime factorization of 420: 2^2 × 3 × 5 × 7.

The sum of 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 + 420 = 1206

Factors can be found using different methods. Mentioned below are some commonly used methods:

• Finding factors using multiplication
   
• Finding factors using the division method
   
• Prime factors and prime factorization

To find factors using multiplication, we need to identify the pairs of numbers that are multiplied to give 420. Identifying the numbers which are multiplied to get the number 420 is the multiplication method.

Step 1: Multiply 420 by 1, 420 × 1 = 420.

Step 2: Check for other numbers that give 420 after multiplying

2 × 210 = 420

3 × 140 = 420

4 × 105 = 420

5 × 84 = 420

6 × 70 = 420

7 × 60 = 420

10 × 42 = 420

12 × 35 = 420

14 × 30 = 420

15 × 28 = 420

20 × 21 = 420

Therefore, the positive factor pairs of 420 are: (1, 420), (2, 210), (3, 140), (4, 105), (5, 84), (6, 70), (7, 60), (10, 42), (12, 35), (14, 30), (15, 28), (20, 21).

For every positive factor, there is a negative factor.

Dividing the given number with whole numbers until the remainder becomes zero and listing out the numbers which result as whole numbers as factors. Factors can be calculated by following the simple division method -

Step 1: Divide 420 by 1, 420 ÷ 1 = 420.

Step 2: Continue dividing 420 by the numbers until the remainder becomes 0.

420 ÷ 1 = 420

420 ÷ 2 = 210

420 ÷ 3 = 140

420 ÷ 4 = 105

420 ÷ 5 = 84

420 ÷ 6 = 70

420 ÷ 7 = 60

420 ÷ 10 = 42

420 ÷ 12 = 35

420 ÷ 14 = 30

420 ÷ 15 = 28

420 ÷ 20 = 21

Therefore, 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, 420.

The factors can be found by dividing it with prime numbers. We can find the prime factors using the following methods:

• Using prime factorization
   
• Using factor tree

Using Prime Factorization: In this process, prime factors of 420 divide the number to break it down in the multiplication form of prime factors till the remainder becomes 1.

420 ÷ 2 = 210 210 ÷ 2 = 105 105 ÷ 3 = 35 35 ÷ 5 = 7 7 ÷ 7 = 1

The prime factors of 420 are 2, 3, 5, and 7.

The prime factorization of 420 is: 2^2 × 3 × 5 × 7.

The factor tree is the graphical representation of breaking down any number into prime factors. The following step shows -

Step 1: Firstly, 420 is divided by 2 to get 210.

Step 2: Now divide 210 by 2 to get 105.

Step 3: Then divide 105 by 3 to get 35.

Step 4: Divide 35 by 5 to get 7. Here, 7 is the smallest prime number that cannot be divided anymore.

So, the prime factorization of 420 is: 2^2 × 3 × 5 × 7.

Factor Pairs Two numbers that are multiplied to give a specific number are called factor pairs.

Both positive and negative factors constitute factor pairs.

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

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

• Factors: The numbers that divide the given number without leaving a remainder are called factors. For example, 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.
   
• Prime factors: The factors which are prime numbers. For example, 2, 3, 5, and 7 are prime factors of 420.
   
• Factor pairs: Two numbers in a pair that are multiplied to give the original number are called factor pairs. For example, the factor pairs of 420 are (1, 420), (2, 210), etc.
   
• Prime factorization: The process of expressing a number as a product of its prime factors. For 420, it's 2^2 × 3 × 5 × 7.
   
• Multiple: A multiple of a number is the product of that number with an integer. For example, 420 is a multiple of 6.