All Factors and Prime Factorization of 420

Understanding the factors of 420 is an essential part of number theory, which helps students master divisibility, prime factorization, and word problems. This concept is crucial not only for school maths exams but also for competitive tests like JEE and in many practical situations involving numbers.

What Are Factors of 420?

A factor of 420 is any integer that divides 420 exactly, leaving no remainder. In other words, if you multiply two whole numbers and get 420, both are factors of 420. Since 420 is a composite number, it has several factors. Understanding factors helps students break down numbers, find common multiples, and solve real-life arithmetic problems.

List of All 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
 
• 420

How to Find Factors of 420

To find the factors of 420, you need to check which numbers divide 420 without leaving a remainder. Here’s a simple step-by-step approach:

 
1. Start with 1 and 420—they always work for any number.
 
2. Divide 420 by 2: result is 210. So, 2 and 210 are factors.
 
3. Try dividing by 3: 420 ÷ 3 = 140. So, 3 and 140 are factors.
 
4. Continue this for all integers up to 420, but you can stop checking once you reach the square root of 420 (approx 20.5), since factors repeat after that (in pairs).

When you find one factor, its pair is always 420 divided by that factor. This systematic checking helps ensure you capture every factor, both small and large.

Prime Factorization of 420

Identifying the prime factors of 420 helps you write it as a product of prime numbers. Here’s how to do it:

 
1. Divide 420 by the smallest prime number, 2: 420 ÷ 2 = 210.
 
2. Divide 210 by 2 again: 210 ÷ 2 = 105.
 
3. Now, 105 is not divisible by 2. Try 3: 105 ÷ 3 = 35.
 
4. Next, 35 is not divisible by 3. Try 5: 35 ÷ 5 = 7.
 
5. 7 is a prime number, so stop here.

So, the prime factorization of 420 is:

420 = 2 × 2 × 3 × 5 × 7

Or in exponential form: 420 = 2² × 3 × 5 × 7

Factor Tree of 420

A factor tree visually breaks a number down into its prime factors. For 420, the factor tree looks like this:

 
• 420 breaks into 2 × 210
 
• 210 breaks into 2 × 105
 
• 105 breaks into 3 × 35
 
• 35 breaks into 5 × 7

Reading from the leaves upwards gives the prime factorization as above.

Factor Pairs of 420

A factor pair is a set of two factors which, when multiplied together, equal 420. Here are all the positive factor pairs of 420:

                                   

Factor 1	Factor 2
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 instance, if you see a word problem asking for “two numbers that multiply to 420 and add up to 41”, you can scan the table above (20 + 21 = 41, so 20 and 21 are the pair).

Worked Examples

Example 1: Find All Odd Factors of 420

 
1. List all factors of 420.
 
2. Pick the ones which are not divisible by 2.
 
3. The odd factors are: 1, 3, 5, 7, 15, 21, 35, 105.

Example 2: Word Problem

You need to arrange 420 books in equal stacks with no books left over. What is the largest number of stacks you can have if each stack must have more than one book, and all stacks must be equal?

 
1. Find the largest factor of 420 less than 420 itself: that's 210.
 
2. Each stack will have 2 books (or use the other large factors depending on the context).

So you can have up to 210 stacks of 2 books each, or other combinations such as 140 stacks of 3 books, etc.

Practice Problems

 
• List all the factors of 420.
 
• Write the prime factorization of 420 in exponential form.
 
• Find all pairs of factors of 420 whose sum is 50.
 
• Is 15 a factor of 420? Show how you know.
 
• What are all the even factors of 420?
 
• Solve: 420 ÷ ? = 7 (find the missing factor).

Common Mistakes to Avoid

 
• Confusing factors (numbers that divide 420) with multiples (numbers that 420 divides).
 
• Leaving out factor pairs with the same number (e.g., if 20 × 21 = 420, list both pairs, as order may matter in some problems).
 
• Not checking divisibility rules systematically (sometimes factors are skipped).
 
• Missing the factor 1—every number is divisible by 1!

Real-World Applications

The concept of factors is widely used in various real-world areas:

 
• Event Planning: Dividing guests or items into equal groups.
 
• LCM/HCF Problems: Used in syncing activities, finding timings (see LCM), and grouping items.
 
• Arrangements: Determining how things can be distributed evenly (like classroom seating).
 
• Problem Solving: Quickly decomposing large numbers for easier multiplication/division tasks in exams or coding contests.

At Vedantu, we help students master such maths concepts for both exams and real-life situations.

Page Summary

In this lesson, we learned what the factors of 420 are, how to find them, factor pairs, and the prime factorization using step-by-step breakdowns and examples. These skills enable students to solve arithmetic, algebraic, and word problems more efficiently. Knowing how to find all the factors of a number is a foundational skill for both academic success and many real-world tasks.