Complete Guide to the Factors of 400

Understanding the factors of 400 is an essential part of arithmetic, especially when dealing with problems in number theory and algebra. Whether you are preparing for your school exams, competitive exams, or simply want to sharpen your Maths skills, learning how to find and apply factors will always be useful. At Vedantu, we make these concepts simple and clear for every student.

What Are the Factors of 400?

A factor of 400 is any whole number that divides 400 exactly, leaving no remainder. In other words, factors are numbers that multiply together in pairs to give the original number, which is 400 here. For example, 5 and 80 are factors because \( 5 \times 80 = 400 \). The complete list of positive factors of 400 is:

• 1
• 2
• 4
• 5
• 8
• 10
• 16
• 20
• 25
• 40
• 50
• 80
• 100
• 200
• 400

Negative factors (like -1, -2, -4, …, -400) also exist since a negative times a negative gives a positive. However, in most cases, we focus on positive factors. The total number of positive factors for 400 is 15.

Understanding Factors, Multiples, and Divisors

• Factors: Numbers that divide 400 completely. Example: 8 is a factor because \( 400 \div 8 = 50 \).
• Multiples: Products you get when you multiply 400 by any whole number (e.g., 400, 800, 1200).
• Divisor: Another name for a factor—any number that divides without leaving a remainder.

For example, 50 is both a divisor and a factor of 400 (\( 400 \div 50 = 8 \)).

Factor Pairs of 400

Factor pairs are two numbers that, when multiplied, result in 400. These pairs include:

Notice the symmetry. For example, (5, 80) and (80, 5) are essentially the same as factor pairs.

Prime Factorization of 400

Prime factorization means writing 400 as a product of only prime numbers. Let's break it down step by step:

1. Divide 400 by 2 (smallest prime): \( 400 \div 2 = 200 \)
2. Divide 200 by 2: \( 200 \div 2 = 100 \)
3. Divide 100 by 2: \( 100 \div 2 = 50 \)
4. Divide 50 by 2: \( 50 \div 2 = 25 \)
5. 25 can't be divided by 2. Try 5: \( 25 \div 5 = 5 \)
6. Finally, \( 5 \div 5 = 1 \)

So, the prime factorization of 400 is:

\( 2 \times 2 \times 2 \times 2 \times 5 \times 5 \) or \( 2^4 \times 5^2 \)

To see this visually, draw a simple factor tree starting with 400, splitting into 2 and 200, and keep breaking down until you reach only prime numbers.

How to Find Factors of 400 Using the Division Method

To find the factors by division:

• Start dividing 400 by 1, 2, 3, 4, ..., up to 400.
• If the division leaves 0 remainder, the number is a factor.

For example, \( 400 \div 8 = 50 \), so both 8 and 50 are factors. \( 400 \div 7 \) leaves a remainder, so 7 is not a factor.

Worked Examples: Factors of 400

Example 1

Find all common factors of 100 and 400.

1. List factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
2. List factors of 400: 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400
3. Common factors are: 1, 2, 4, 5, 10, 20, 25, 50, 100

Example 2

Rachel wants to find the sum of all factors of 400.

• Add all: 1 + 2 + 4 + 5 + 8 + 10 + 16 + 20 + 25 + 40 + 50 + 80 + 100 + 200 + 400 = 961

Example 3

Is 16 a factor of 400?

• \( 400 \div 16 = 25 \), so yes, 16 is a factor.

Practice Problems

• List all the factors of 200.
• Find all factor pairs of 400 whose sum is 41.
• What is the greatest prime factor of 400?
• Is 8 a factor or a multiple of 400?
• Write the prime factorization of 400 using exponents.
• Find two factors of 400 that add up to 40.
• Create a factor tree for 400.

Common Mistakes to Avoid

• Confusing factors with multiples. (Factors divide the number; multiples are results of multiplying the number.)
• Forgetting to include 1 and the number itself as factors.
• Missing repeated factors in factor pairs (e.g., 20 × 20 = 400).
• Assuming every divisor is a prime factor (prime factorization uses only primes).

Real-World Applications

Factors are used everywhere: sharing objects equally, dividing land into plots, or calculating areas (like making a rectangular garden with perimeter 400 meters). In competitive exams like JEE and NEET, factorization techniques help solve LCM, HCF, and divisibility questions. Factor pairs can also help with grouping and arranging in games and business settings.

At Vedantu, we teach such number properties through interactive lessons and quizzes, so learning factors becomes easy and fun!

In summary, knowing the factors of 400 helps you solve number problems, practice arithmetic, and understand properties of numbers. Remember the factor list, explore prime factorization, and avoid common mistakes. Practice more with Vedantu to master factors and excel in Maths!