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Factors of 420

420 is the sum of the first 20 even numbers. It is the smallest number that is divisible by numbers 1 to 7. In this lesson, let us calculate the factors of 420, prime factors of 420, and factors of 420 in pairs along with solved examples for a better understanding.

  • 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 : 2 × 2 × 3 × 5 × 7 = 22 × 3 × 5 × 7
1. What Are the Factors of 420?
2. How to Calculate Factors of 420?
3. Factors of 420 by Prime Factorization
4. Factors of 420 in Pairs
5. Important Notes
6. Challenging Questions
7. FAQs on Factors of 420

What are the Factors of 420?

All the numbers that divide 420 completely, leaving no remainder are the factors of 420. 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.

How to Calculate the Factors of 420?

Let's learn how to calculate factors of 420. 420 is divisible by 1 and itself. Let's check the divisibililty of 420 with the other numbers, starting from 2 and obtain the factors in pairs. let's use the divisibility rules and check for divisibility. 420 is not divisible by 8, 9, 13, 16, 17, 18, 19, 22 and so on. In all the cases below, 420 ÷ Divisor = Quotient, where remainder = 0 and also observe that 420 ÷ Quotient = Divisor

DIVISOR QUOTIENT
2 210
3 140
4 105
5 84
6 70
7 60
10 42
12 35
14 30
15 28
20 21

Explore factors using illustrations and interactive examples

  • Factors of 512: The factors of 512 are 1, 2, 4, 8, 16, 32, 64, 128, 256, and 512.
  • Factors of 175: The factors of 175 are 1, 5, 7, 25, 35, 175.
  • Factors of 216: The factors of 216 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108 and 216
  • Factors of 112: The factors of 112 are  1, 2, 4, 7, 8, 14, 16, 28, 56, and 112.

Factors of 420 by Prime Factorization

Factors of 420 by prime factorization are given by using the following steps. We determine the prime factors using the division method or the factor-tree method. 

  • Find the smallest prime number by which 420 can be divided without leaving any remainder.
  • Express the prime factor as a branch and its composite pair factor on the other branch. Here, we have 2 × 210 as our branches.
  • Again 210 is further as the product of its smallest prime factor 2 and the composite pair factor 105.
  • Keep branching out the tree until we obtain the prime factors at the end of the branches.

prime factorization of 420 to find the prime factors of 420

  • This way we evaluate all the prime factors and express 420 as a product of its prime factors. Prime factorization of 420 = 2 × 2 × 3 × 5 × 7 = 22 × 3 × 5 × 7.
  • The prime factors of 420 are 2 , 3 , 5 and 7
  • Consider the exponents alone of these prime factors. Add one to each of these exponents and find the product of them.
  • (2 +1) × (1 + 1)  × (1 + 1)  × (1 + 1) = 3 × 2 × 2 × 2 = 24 factors 
  • This helps us in determining the total number of factors of 420.

Factors of 420 in Pairs

The pair of numbers when multipled together gets the number 420. Number 1 × Number 2 = 420. This is expressed as factors of 420 in ordered pairs as (number 1, number2). We have 12 such factor-pairs of 420 as observed in the pattern below. The numbers in the top row multiplied with the numbers in the bottom row form the factors of 420 in pairs.

factors of 420 in pairs

The distinct 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),(14,30), (15,28) and (20,21)
The factor pairs can be negative also.
-ve × -ve = +ve
The negative 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),(-14, -30), (-15, -28) and (-20, -21)

Important Notes

  • There are 24 factors of 420 in all.
  • 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 that divide 420 evenly without any remainder.
  • The prime factors of 420 are 2, 3, 5 and 7.

Challenging Questions

  • 420 = 22 × 3 × 5 × 7 and 320 = 26 × 5. Find the greatest common factor among these.
  • Evaluate taking the common factors out: (210 × 105) + (420 × 140)
 

Factors of 420 Solved Examples

  1. Example 1: George is a grocer and he has arranged them in rows in his stall. He has 420 apples, oranges and strawberries each. How many fruits are there in each row if he has arranged apples in 28 rows, oranges in 35 rows and strawberries in 20 rows?

    Solution:

    420 apples are arranged in 28 rows is written as : ___ apples × 28 rows = 420 apples
    ___ × 28 = 420. Finding the missing factor-pair,  15 × 28 = 420
    420 oranges are arranged in 35 rows :___ oranges × 35 rows = 420 oranges
    ___ × 35 = 420. Finding the missing factor-pair,  12 × 35 = 420
    420 strawberries are arranged in 20 rows:___ apples × 28 rows = 420 strawberries
    ___ × 20 = 420. Finding the missing factor-pair,  21 × 20 = 420
    Thus we determine that George has arranged 15 apples in 28 rows, 12 apples in 35 rows and 21 strawberries in 20 rows.

  2. Example 2: Shawn travels 420 miles in 30 hours. What could be the distance traveled by him in an hour?

    Solution:

    We know that total miles covered is the product of distance covered in one hour × total time taken 
    ___ × 30 hours = 420 miles
    Finding the missing factor, we get 14  × 30 = 420 miles
    Thus he has travelled 14 miles in an hour.

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FAQs on Factors of 420

What are 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. They divide 420 evenly, leaving no remainder.

What are the common factors of 420 and 400?

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.
The factors of 400 are 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200 and 400.
The common factors of  1, 2, 4, 5, 10 and 20

What are the factors of 420 that add up to 29?

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.
Among these 14 and 15 sum up to 29.

What are the total number of factors of 420?

There are 24 factors of 420 in all. 

What are the prime factors of 420?

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

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