What Are the Factor Pairs of 42?
The concept of factors of 42 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Learning about factors helps you in arithmetic, algebra, divisibility, and many competitive exams. This page will help you easily find all factors of 42, understand their uses, and master speedy techniques for problem-solving.
What Is Factors of 42?
A factor of 42 is any whole number that divides 42 exactly with no remainder. You’ll find this concept applied in areas such as LCM & HCF, number theory, and divisibility rules. Factors are building blocks for understanding prime numbers, multiples, and even algebraic terms. Here’s the list of all factors of 42: 1, 2, 3, 6, 7, 14, 21, 42.
Key Formula for Factors of 42
There’s no single formula to find all factors, but here’s the general process:
If N is a number, all integers "d" where N ÷ d has no remainder are factors of N. For 42, check numbers from 1 to 42.
Cross-Disciplinary Usage
Factors of 42 are not only useful in Maths but also play an important role in Physics (for ratios, gear systems), Computer Science (programming logic, cryptography), and daily logical reasoning. Students preparing for Olympiads, JEE, or NEET will see its relevance in several problem types and MCQs.
Step-by-Step Illustration
- List all numbers between 1 and 42.
- Divide 42 by each number. If the division gives a whole number, that’s a factor.
- Record the results:
- Final factor list: 1, 2, 3, 6, 7, 14, 21, 42
42 ÷ 2 = 21 (factor: 2)
42 ÷ 3 = 14 (factor: 3)
42 ÷ 6 = 7 (factor: 6)
42 ÷ 7 = 6 (factor: 7)
42 ÷ 14 = 3 (factor: 14)
42 ÷ 21 = 2 (factor: 21)
42 ÷ 42 = 1 (factor: 42)
Visualization: Factor Pairs of 42
| Pair # | Factor 1 | Factor 2 |
|---|---|---|
| 1 | 1 | 42 |
| 2 | 2 | 21 |
| 3 | 3 | 14 |
| 4 | 6 | 7 |
Prime Factorization and Factor Tree of 42
Prime factors are the basic building blocks of a number. The prime factorization of 42 is breaking it down to the product of primes:
- Divide 42 by 2: 42 ÷ 2 = 21
- Divide 21 by 3: 21 ÷ 3 = 7
- Divide 7 by 7: 7 ÷ 7 = 1
- Prime factors: 2, 3, 7
A factor tree for 42 splits into 2 × 21, then 21 into 3 × 7. So, 42 = 2 × 3 × 7.
Quick Chart: Summary for 42
| All Factors | # Positive Divisors | Even Factors | Odd Factors |
|---|---|---|---|
| 1, 2, 3, 6, 7, 14, 21, 42 | 8 | 2, 6, 14, 42 | 1, 3, 7, 21 |
Speed Trick or Vedic Shortcut
If you need to quickly check if a number is a factor of 42, see if multiplying it by anything gives 42 or divide 42 by the number and check for remainders. Also, you can remember the prime factor set (2 × 3 × 7) and build any factor from their combinations.
Example Trick: To quickly list factors, start with 1 and keep pairing up so 1×42, 2×21, 3×14, 6×7. Stop pairing when both numbers are equal or overlap.
- Find the smallest factor pair (start with 1): 1 × 42
- Go up: 2 × 21, 3 × 14, 6 × 7
- Stop when factors repeat in pairs
- You have found all!
Such Vedic math tricks are taught in Vedantu’s live classes for exam speed.
Try These Yourself
- Write all factors of 42 in ascending order.
- Check if 28 is a factor of 42 (Try: 42 ÷ 28?).
- List all even factors of 42.
- Find all positive factor pairs of 42.
- Find the sum of all factors of 42.
Frequent Errors and Misunderstandings
- Confusing factors and multiples. (Factors divide the number; multiples are the results of multiplication.)
- Missing pairs (for example, forgetting 2 × 21).
- Mixing up factors with only prime factors.
Relation to Other Concepts
The idea of factors of 42 connects closely with topics such as prime factorization and LCM & HCF. Mastering this helps with finding highest common factors, lowest common multiples, and simplifies many algebraic solutions in later chapters.
Classroom Tip
A quick way to remember factors is to write them as pairs on paper, or use factor trees for visualization. Vedantu’s teachers often use colors or charts in live classes to help you visualize and remember paired factors.
We explored factors of 42—from definition, lists, step-by-step checks, visuals, and classic mistakes to relationships with other topics. Keep practicing factor questions with solved examples and quiz yourself using the tricks above. Continue your learning on Vedantu to boost your Maths confidence!
Related Resources:
- Prime Factors of 42: See how 42 breaks into multiples of prime numbers.
- Factors of 60: Practice finding factors of related numbers and compare similarities.
- LCM & HCF: Learn how factors help in solving LCM/HCF problems.
- Factorization Methods: Deepen your techniques for breaking numbers like 42 down.
FAQs on Factors of 42 Explained with Examples
1. What are the factors of 42?
The factors of 42 are the whole numbers that divide 42 exactly without leaving a remainder. These are: 1, 2, 3, 6, 7, 14, 21, and 42. These numbers, when multiplied in pairs, result in 42.
2. How do you find all factor pairs of 42?
To find all factor pairs of 42, systematically pair numbers that multiply to 42:
• (1, 42)
• (2, 21)
• (3, 14)
• (6, 7)
These pairs represent all the possible combinations of two factors that yield a product of 42.
3. What is the prime factorization of 42?
The prime factorization of 42 is 2 x 3 x 7. This means 42 can be expressed as the product of only prime numbers (numbers divisible only by 1 and themselves).
4. What are the prime factors of 42?
The prime factors of 42 are 2, 3, and 7. These are the prime numbers that, when multiplied together, equal 42.
5. How can I use a factor tree to find the factors of 42?
A factor tree visually represents the prime factorization. Start with 42. Branch it into two factors (e.g., 2 and 21). Continue branching until all branches end in prime numbers. The prime factors at the end of the branches (2, 3, and 7) are the prime factorization of 42. From there, you can derive all factors.
6. Are all factors of 42 also multiples of 42?
No. Factors are numbers that divide 42 evenly, while multiples are numbers that are obtained by multiplying 42 by other whole numbers. For example, 1, 2, 3, 6, 7, 14, 21, and 42 are factors, but 84, 126, etc. are multiples.
7. Can 42 be divided by 6 and 7?
Yes. Both 6 and 7 are factors of 42. 42 ÷ 6 = 7 and 42 ÷ 7 = 6.
8. Is 42 a prime number?
No. 42 is a composite number because it has factors other than 1 and itself.
9. What are some real-world applications of finding factors?
Finding factors is useful in many areas, including:
• Simplifying fractions: Finding common factors helps reduce fractions to their simplest form.
• Solving algebraic equations: Factoring expressions is crucial in solving various algebraic problems.
• Calculating the Least Common Multiple (LCM) and Highest Common Factor (HCF): Understanding factors is essential for determining LCM and HCF, which are frequently used in various mathematical applications.
10. How do negative numbers work as factors of 42?
Negative numbers can also be factors. A negative number multiplied by another negative number results in a positive product. Thus, -1, -2, -3, -6, -7, -14, -21, and -42 are also factors of 42.
11. What is the difference between factors and divisors?
The terms 'factors' and 'divisors' are often used interchangeably in mathematics, referring to numbers that divide a given number exactly. In the context of 42, both terms denote the numbers 1, 2, 3, 6, 7, 14, 21, and 42 (and their negative counterparts).
