Factors of 840

The numbers that divide 840 evenly are known as factors of 840.

A factor of 840 is a number that divides the number without a remainder.

The factors of 840 are 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, and 840.

Negative factors of 840: -1, -2, -3, -4, -5, -6, -7, -8, -10, -12, -14, -15, -20, -21, -24, -28, -30, -35, -40, -42, -56, -60, -70, -84, -105, -120, -140, -168, -210, -280, -420, and -840.

Prime factors of 840: 2, 3, 5, and 7.

Prime factorization of 840: (2³ x 3 x 5 x 7).

 The sum of factors of 840: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 10 + 12 + 14 + 15 + 20 + 21 + 24 + 28 + 30 + 35 + 40 + 42 + 56 + 60 + 70 + 84 + 105 + 120 + 140 + 168 + 210 + 280 + 420 + 840 = 2,232

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

1. Finding factors using multiplication
    
2. Finding factors using the division method
    
3. Prime factors and prime factorization

To find factors using multiplication, we need to identify the pairs of numbers that are multiplied to give 840. Identifying the numbers which are multiplied to get the number 840 is the multiplication method.

Step 1: Multiply 840 by 1, 840 × 1 = 840.

Step 2: Check for other numbers that give 840 after multiplying

(2 x 420 = 840) 

(3 x 280 = 840)

(4 x 210 = 840) 

(5 x 168 = 840) 

(6 x 140 = 840) 

(7 x 120 = 840) 

(8 x 105 = 840) 

(10 x 84 = 840) 

(12 x 70 = 840)

(14 x 60 = 840) 

(15 x 56 = 840) 

(20 x 42 = 840) 

(21 x 40 = 840) 

(24 x 35 = 840) 

(28 x 30 = 840)

Therefore, the positive factor pairs of 840 are: (1, 840), (2, 420), (3, 280), (4, 210), (5, 168), (6, 140), (7, 120), (8, 105), (10, 84), (12, 70), (14, 60), (15, 56), (20, 42), (21, 40), (24, 35), and (28, 30).

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 840 by 1, 840 ÷ 1 = 840.

Step 2: Continue dividing 840 by the numbers until the remainder becomes 0.

840 ÷ 1 = 840

840 ÷ 2 = 420

840 ÷ 3 = 280

840 ÷ 4 = 210

840 ÷ 5 = 168

840 ÷ 6 = 140

840 ÷ 7 = 120

840 ÷ 8 = 105

840 ÷ 10 = 84

840 ÷ 12 = 70

840 ÷ 14 = 60

840 ÷ 15 = 56

840 ÷ 20 = 42

840 ÷ 21 = 40

840 ÷ 24 = 35

840 ÷ 28 = 30

Therefore, the factors of 840 are: 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, and 840.

The factors can be found by dividing 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 840 divide the number to break it down in the multiplication form of prime factors till the remainder becomes 1.

840 ÷ 2 = 420

420 ÷ 2 = 210

210 ÷ 2 = 105

105 ÷ 3 = 35

35 ÷ 5 = 7

7 ÷ 7 = 1

The prime factors of 840 are 2, 3, 5, and 7.

The prime factorization of 840 is: (2³ x 3 x 5 x 7).

The factor tree is the graphical representation of breaking down any number into prime factors. The following steps show:

Step 1: Firstly, 840 is divided by 2 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 840 is: (2³ x 3 x 5 x 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 840: (1, 840), (2, 420), (3, 280), (4, 210), (5, 168), (6, 140), (7, 120), (8, 105), (10, 84), (12, 70), (14, 60), (15, 56), (20, 42), (21, 40), (24, 35), and (28, 30).

• Negative factor pairs of 840: (-1, -840), (-2, -420), (-3, -280), (-4, -210), (-5, -168), (-6, -140), (-7, -120), (-8, -105), (-10, -84), (-12, -70), (-14, -60), (-15, -56), (-20, -42), (-21, -40), (-24, -35), and (-28, -30).

• Factors: The numbers that divide the given number without leaving a remainder are called factors. For example, the factors of 840 are 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, and 840.

• Prime factors: The factors which are prime numbers. For example, 2, 3, 5, and 7 are prime factors of 840.

• 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 840 are (1, 840), (2, 420), and so on.

• Prime factorization: The method of expressing a number as the product of its prime factors. For example, the prime factorization of 840 is (2³ x 3 x 5 x 7).

• Division method: A way to find factors by dividing the number by natural numbers to see which divisions result in a whole number.