Factors of 126 Complete Guide with Definition and Methods

Understanding the factors of 126 is a foundational part of arithmetic and number theory, important for solving school Maths problems and preparing for competitive exams like JEE and NEET. Learning how to identify and use factors makes calculations easier and can help with other Maths concepts such as HCF, LCM, and divisibility rules.

What are Factors of 126?

A factor of a number is a whole number that divides the number exactly, leaving no remainder. The factors of 126 are all the positive and negative integers which can be multiplied in pairs to result in 126. These factors show how the number 126 can be broken down and are essential in both basic and advanced Mathematics.

How to Find the Factors of 126?

To find the factors of 126, we check which numbers can divide 126 exactly (without leaving a remainder). Begin with 1 and go up to 126:

• 126 ÷ 1 = 126  → 1 and 126 are factors
• 126 ÷ 2 = 63  → 2 and 63 are factors
• 126 ÷ 3 = 42  → 3 and 42 are factors
• 126 ÷ 6 = 21  → 6 and 21 are factors
• 126 ÷ 7 = 18  → 7 and 18 are factors
• 126 ÷ 9 = 14  → 9 and 14 are factors

By continuing this method, we find all divisors. Therefore, the factors of 126 are:

1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126

For a general method, divide 126 by each number from 1 up to its square root (about 11.23). Every time it divides evenly, both the divisor and the quotient are factors.

Pair Factors of 126

Pair factors are two numbers that multiply together to give 126. These pairs are useful for understanding multiplication and area problems.

Both positive and negative pairs multiply to 126, since the product of two negative numbers is positive. Pair factors often help when solving problems involving multiplying numbers to get a certain product.

Prime Factorization of 126

Prime factorization breaks 126 into its smallest prime number components. This is important for finding HCF, LCM, and Euler’s Totient function.

• Start with the smallest prime (2): 126 ÷ 2 = 63
• Next prime (3): 63 ÷ 3 = 21
• Continue with 3: 21 ÷ 3 = 7
• And 7 is itself prime: 7 ÷ 7 = 1

So the prime factorization of 126 is:

126 = 2 × 3 × 3 × 7 (or 2 × 3² × 7)

Prime factors of 126: 2, 3, 7

To learn more about prime numbers and explore all methods, visit Vedantu’s lesson on Prime Factorization.

Worked Examples

1. What number should be multiplied by 21 to get 126?

   • x × 21 = 126
   • x = 126/21 = 6
   • Answer: 6

2. What is the sum of all factors of 126?

   • Sum = 1 + 2 + 3 + 6 + 7 + 9 + 14 + 18 + 21 + 42 + 63 + 126 = 312
3. Answer: 312

4. What is the greatest common factor (GCF) of 120 and 126?

   • Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
   • Factors of 126: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126
   • Common factors: 1, 2, 3, 6
     GCF: 6

For more examples and interactive videos, visit our lesson on How to Find Factors of a Number on Vedantu.

Practice Problems

• List all the factors of 126.
• Find all the pair factors (positive and negative) of 126 and write them.
• What is the product of the greatest and the smallest factor of 126?
• Find the prime factorization of 126 using a factor tree.
• Is 18 a factor of 126?
• If 126 = 2 × 3 × 3 × 7, what is the sum of its unique prime factors?
• Find the common factors of 126 and 144.

For more practice, try solving problems on Factors of 120 and Factors of 105 to compare results!

Common Mistakes to Avoid

• Confusing factors with multiples (multiples of 126 are 126, 252, ... but factors are numbers that divide 126).
• Missing pairs of factors by stopping at the wrong point (always check up to the square root).
• Forgetting to include 1 and the number itself as factors.
• Leaving out negative pair factors for completeness, especially in higher classes.
• Not listing repeated prime factors correctly (3 × 3 instead of just 3) in prime factorization.

Real-World Applications

Knowing the factors of 126 and factorization helps in dividing things into groups, like arranging 126 students equally in teams, packaging 126 items, or solving algebraic equations. Factorization is also vital in coding, cryptography, and understanding probability in real life. Many business and tech calculations rely on concepts of factors, LCM, and HCF.

For more on number properties, check the Number System page on Vedantu.

In this topic, we explored the factors of 126, learned step-by-step methods to find them, listed pair and prime factors, and reviewed important examples. Mastering factors is a core Maths skill that helps you solve problems quickly and gives you the confidence to tackle related topics in school and competitive exams. Keep practicing with Vedantu to deepen your understanding and sharpen your Maths skills!