Step-by-Step Guide to Prime and Pair Factors of 126
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.
| Positive Pair Factors | Negative Pair Factors | Product |
|---|---|---|
| 1, 126 | -1, -126 | 1 × 126 = 126 |
| 2, 63 | -2, -63 | 2 × 63 = 126 |
| 3, 42 | -3, -42 | 3 × 42 = 126 |
| 6, 21 | -6, -21 | 6 × 21 = 126 |
| 7, 18 | -7, -18 | 7 × 18 = 126 |
| 9, 14 | -9, -14 | 9 × 14 = 126 |
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
-
What number should be multiplied by 21 to get 126?
- x × 21 = 126
- x = 126/21 = 6
- Answer: 6
-
What is the sum of all factors of 126?
- Sum = 1 + 2 + 3 + 6 + 7 + 9 + 14 + 18 + 21 + 42 + 63 + 126 = 312
- Answer: 312
-
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!
FAQs on What Are the Factors of 126?
1. What are the factors of 126?
The factors of 126 are the whole numbers that divide 126 without leaving a remainder. They are: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, and 126. These numbers, when multiplied by another whole number, equal 126.
2. What is the prime factorization of 126?
The prime factorization of 126 breaks it down into its prime factors. It is expressed as 2 x 3 x 3 x 7 or 2 x 32 x 7. This means 126 is a product of these prime numbers only.
3. What are the pair factors of 126?
Pair factors of 126 are pairs of numbers that multiply to give 126. Some examples include (1, 126), (2, 63), (3, 42), (6, 21), (7, 18), (9, 14). Remember that negative pairs are also factors (e.g., (-1, -126)).
4. How do I find all the factors of 126?
To find all factors, systematically divide 126 by each whole number starting from 1. If the division results in a whole number, that number is a factor. Continue this process until you reach the square root of 126 (approximately 11.2). Any factors found beyond this point will be repetitions of already identified factors.
5. What is the difference between factors and multiples of 126?
Factors are numbers that divide 126 evenly, while multiples are numbers that are the result of multiplying 126 by other whole numbers. For example, factors of 126 include 1, 2, 3, etc., while multiples are 126, 252, 378, etc.
6. Is 126 a prime or composite number?
126 is a composite number because it has factors other than 1 and itself (it is divisible by 2, 3, 6, 7, 9, 14 etc.). A prime number has only two factors: 1 and the number itself.
7. How do I use a factor tree to find the prime factors of 126?
A factor tree visually represents the prime factorization. Start with 126. Break it down into two factors (e.g., 2 and 63). Continue breaking down composite factors until you only have prime numbers. For 126, the tree would show branches leading to 2, 3, 3, and 7.
8. What are some real-world applications of finding factors?
Finding factors is used in various areas like simplifying fractions, solving algebraic equations, determining divisibility, and understanding ratios and proportions. It's fundamental in many mathematical concepts.
9. What are the factors of 126 that add up to 25?
The factors of 126 that add up to 25 are 9 and 16. Note that 16 is not a factor of 126, showing that not every sum of factors will produce a factor of the number itself.
10. Explain how to find the greatest common factor (GCF) of 126 and another number, say 18?
To find the greatest common factor (GCF) of 126 and 18, list the factors of each number. Then identify the largest factor they share. For 126, factors include 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126. For 18, factors are 1, 2, 3, 6, 9, 18. The GCF of 126 and 18 is 18.
11. What are the negative factors of 126?
Since multiplying two negative numbers results in a positive number, all the positive factors of 126 also have negative counterparts. So, the negative factors of 126 are -1, -2, -3, -6, -7, -9, -14, -18, -21, -42, -63, and -126.
