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 remainder.

The positive 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: -1 × 2² × 3 × 5 × 7.

The sum of the positive 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 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 that 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 numbers with whole numbers until the remainder becomes zero and listing out the numbers which result in 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 them 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 ÷ -1 = 420

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: -1 × 2² × 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 -1 to get 420.

Step 2: Now divide 420 by 2 to get 210.

Step 3: Then divide 210 by 2 to get 105.

Step 4: Divide 105 by 3 to get 35.

Step 5: 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: -1 × 2² × 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.

• Negative factors: Factors that are negative integers. For example, -1, -2, -3, etc., are negative factors of -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 positive factor pairs of -420 are (1, -420), (2, -210), etc.

• Prime factorization: The expression of a number as a product of its prime numbers. For example, the prime factorization of -420 is -1 × 2² × 3 × 5 × 7.