What Are the Factors and Factor Pairs of 140?
The concept of Factors of 140 is very important in arithmetic and number theory. Understanding factors not only builds a strong foundation for school mathematics but also helps students master more advanced topics like LCM, HCF, and prime factorization. This basic yet powerful idea often comes up in competitive exams and everyday problem-solving.
Understanding Factors of 140
A factor of 140 is any whole number that divides 140 exactly without leaving any remainder. In other words, when 140 is divided by its factor, the answer is always a whole number. Identifying factors helps in simplifying math problems, finding common divisors, and solving questions involving arrays or equal groups.
It is important to remember that factors are different from multiples. Factors are numbers that can be multiplied together to produce 140, while multiples are numbers you get by multiplying 140 by another integer.
What Are the Factors of 140?
Let’s find all the numbers that divide 140 completely. Start by dividing 140 by every whole number from 1 up to 140:
| Factor | 140 ÷ Factor | Result (No Remainder?) |
|---|---|---|
| 1 | 140 ÷ 1 | 140 (Yes) |
| 2 | 140 ÷ 2 | 70 (Yes) |
| 4 | 140 ÷ 4 | 35 (Yes) |
| 5 | 140 ÷ 5 | 28 (Yes) |
| 7 | 140 ÷ 7 | 20 (Yes) |
| 10 | 140 ÷ 10 | 14 (Yes) |
| 14 | 140 ÷ 14 | 10 (Yes) |
| 20 | 140 ÷ 20 | 7 (Yes) |
| 28 | 140 ÷ 28 | 5 (Yes) |
| 35 | 140 ÷ 35 | 4 (Yes) |
| 70 | 140 ÷ 70 | 2 (Yes) |
| 140 | 140 ÷ 140 | 1 (Yes) |
So, the factors of 140 are: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140.
Prime Factorization of 140
The prime factorization of a number means expressing it as a product of only prime numbers. For 140, the process looks like this:
- Divide 140 by 2 (smallest prime): 140 ÷ 2 = 70
- Divide 70 by 2 again: 70 ÷ 2 = 35
- Divide 35 by 5: 35 ÷ 5 = 7
- 7 is already a prime number.
So, the prime factorization of 140 is 2 × 2 × 5 × 7 or 22 × 5 × 7.
Here’s the factor tree representation for better visualization:
- 140
- 2 × 70
- 2 × 35
- 5 × 7
Factor Pairs of 140
A factor pair of 140 is a set of two numbers that, when multiplied together, give 140. Here are all the factor pairs:
| Positive Factor Pair | Negative Factor Pair |
|---|---|
| 1 × 140 | -1 × -140 |
| 2 × 70 | -2 × -70 |
| 4 × 35 | -4 × -35 |
| 5 × 28 | -5 × -28 |
| 7 × 20 | -7 × -20 |
| 10 × 14 | -10 × -14 |
Both numbers in a pair are factors of 140. Changing their order (ex: 140 × 1) gives the same product.
Properties of Factors of 140
- Every factor of 140 is also a divisor of 140.
- 140 has 12 positive factors.
- All even factors of 140: 2, 4, 10, 14, 20, 28, 70, 140.
- All odd factors of 140: 1, 5, 7, 35.
- The sum of all positive factors is 336.
- 8 is not a factor, because 140 ÷ 8 is not a whole number.
- 140 is a composite number because it has factors other than 1 and itself.
How to Find Factors of Any Number: Step-by-Step
- Start with 1 and check if it divides the number exactly.
- Try each consecutive number up to the square root of that number.
- If division is exact, include both the divisor and the quotient as factors.
- Continue until all factor pairs are found.
For more help with factorization, check out Factors of a Number on Vedantu.
Worked Examples
Example 1: List all the factors of 140.
- Start dividing 140 by whole numbers from 1 upward.
- If the result is a whole number, keep that number as a factor.
- Continue until 140 ÷ 140.
The factors of 140 are: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140.
Example 2: Find the common factors of 70 and 140.
Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70.
Factors of 140: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140.
So, the common factors are: 1, 2, 5, 7, 10, 14, 35, 70.
Example 3: Express 140 as a product of its prime factors.
From the factor tree: 140 = 2 × 2 × 5 × 7 or 22 × 5 × 7.
Practice Problems
- Find all factors of 20. Which ones are also factors of 140?
- List all factor pairs of 140 that add up to more than 50.
- Is 8 a factor of 140? Explain your answer.
- Express 140 as a product of three different numbers (not including 1 or 140).
- What are the first three multiples of 140?
Common Mistakes to Avoid
- Confusing factors with multiples. Remember, factors divide numbers exactly; multiples are results of multiplying the number with other integers.
- Forgetting pairs: Each factor less than the square root pairs with a greater factor.
- Missing out on 1 and the number itself as factors.
- Including non-integer results (like 140 ÷ 8 = 17.5) as factors. Only whole numbers count!
Real-World Applications
Factors are used in real life whenever we need to make groups of items. For example, if you have 140 chocolates and want to distribute them equally, knowing the factors of 140 helps you know in how many ways you can divide them without any remainder. Concepts like these are applied in dividing seats, resources, and even in computer programming and cryptography.
At Vedantu, we simplify concepts like factors of 140 to help students build strong basics for success in exams and in life’s daily math problems.
To practice more, check out related topics like Factors of 60, Prime Numbers, or LCM and HCF on Vedantu.
In summary, understanding the factors of 140 gives you tools for breaking down numbers, solving common math problems, and exploring advanced topics. Practicing finding factors makes calculations faster and prepares you well for school and competitive exams.
FAQs on Factors of 140 Explained with Examples
1. What are the factors of 140?
The factors of 140 are the numbers that divide 140 without leaving a remainder. They are: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Finding factors is crucial in number theory and solving various math problems.
2. What is a factor tree of 140?
A factor tree visually represents the prime factorization of a number. For 140, you would break it down into its prime factors: 2, 5, and 7. A typical tree would show 140 branching into 14 and 10, then further branching down to the prime factors. This helps visualize how prime factors multiply to reach the original number.
3. What is 140 divisible by?
140 is divisible by all its factors: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Divisibility rules help determine if a number is divisible by certain factors quickly, saving time in calculations and problem-solving.
4. Is 8 a factor of 140?
No, 8 is not a factor of 140. 140 divided by 8 leaves a remainder; therefore, 8 does not divide 140 exactly. Understanding divisibility rules assists in quickly determining factors.
5. What are the factors of 140 in pairs?
The factor pairs of 140 are pairs of numbers that multiply to give 140. These include: (1, 140), (2, 70), (4, 35), (5, 28), (7, 20), (10, 14). Understanding factor pairs is useful for various mathematical operations.
6. Factors of 140 that add up to 31?
To find factors of 140 that add up to 31, you would need to check all factor pairs. However, none of the factor pairs of 140 add up to 31.
7. Factors of 140 that add up to 24?
The factors 14 and 10 add up to 24 and their product is 140. Finding factors that meet a specific sum is a common problem type in number theory and applications.
8. Factors of 140 by division method?
The division method involves systematically dividing 140 by numbers starting from 1, and recording the pairs of factors obtained from each division. Keep dividing until the quotient becomes smaller than the divisor. The resulting pairs represent all the factors of 140.
9. How do you find the prime factors of 140?
To find the prime factors of 140, use the factor tree method or repeated division by prime numbers (2, 3, 5, 7, etc.). This will give you the prime factorization, showing the prime numbers that multiply to 140 (2 x 2 x 5 x 7). Prime factorization is fundamental in number theory.
10. How many factors does 140 have?
140 has a total of 12 factors. These are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. The number of factors can be determined from its prime factorization.
11. What are the prime factors of 140?
The prime factors of 140 are 2, 5, and 7. These are prime numbers that, when multiplied together, result in 140. Prime factorization is used extensively in higher-level mathematics.
