What are the Factor Pairs and Prime Factorization of 74?
The concept of factors of 74 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding factors helps students solve problems related to divisibility, HCF, LCM, and simplifies arithmetic operations during board exams and competitive tests.
What Are the Factors of 74?
A factor of 74 is any whole number that divides 74 exactly, leaving no remainder. In other words, if 74 divided by a number results in another whole number, that number is a factor of 74. Students use this concept in topics like HCF (Highest Common Factor), LCM (Lowest Common Multiple), and while learning about prime numbers, divisibility, and multiples.
List of Factors of 74
Factors of 74: 1, 2, 37, and 74. Each number divides 74 without leaving any remainder. Remember, every whole number has both positive and negative factors, but in most school exams, only positive factors are listed unless stated otherwise.
- 1 × 74 = 74
- 2 × 37 = 74
How to Find Factors of 74 (Division and Pair Method)
To calculate the factors of 74, check which whole numbers divide 74 without leaving a remainder. Use the following step-by-step process for quick exams or homework:
- Start with 1: 74 ÷ 1 = 74 (No remainder)
- Next, try 2: 74 ÷ 2 = 37 (No remainder)
- Try 3: 74 ÷ 3 ≈ 24.67 (Remainder exists, so not a factor)
- Continue up to 74. The numbers that exactly divide 74 are:
1, 2, 37, and 74
These are the only factors of 74 because no other whole numbers divide 74 exactly.
Factor Pairs of 74
Factors can also be written as pairs that multiply to give 74. For example, (1, 74) means 1 × 74 = 74. Here is a complete table of factor pairs:
| Factor Pair | Multiplication |
|---|---|
| (1, 74) | 1 × 74 = 74 |
| (2, 37) | 2 × 37 = 74 |
Negative Factors of 74
Every positive factor also has a corresponding negative factor. For example, (-1) × (-74) = 74, and (-2) × (-37) = 74. So the negative factors are -1, -2, -37, and -74.
Prime Factorization of 74
Prime factorization breaks a number down to just its prime number factors. For 74, start with the smallest prime:
1. 74 ÷ 2 = 37 (2 is a prime number.)2. 37 ÷ 37 = 1 (37 is prime, so stop.)
Prime factorization of 74: 74 = 2 × 37. Both 2 and 37 are prime numbers. You can draw a simple factor tree starting with 74 at the top, splitting into 2 and 37 at the branches.
Is 74 a Prime or Composite Number?
74 is a composite number because it has more than two factors (1, 2, 37, and 74). Prime numbers have only two factors: 1 and themselves.
Multiples of 74
Multiples of 74 are numbers you get by multiplying 74 with any whole number. Here are the first five multiples:
- 74 × 1 = 74
- 74 × 2 = 148
- 74 × 3 = 222
- 74 × 4 = 296
- 74 × 5 = 370
Solved Examples: Factors of 74
1. List all factors of 74.
Solution: 1, 2, 37, and 74.
2. What is the sum of all positive factors of 74?
Solution: 1 + 2 + 37 + 74 = 114.
3. Find the common factors of 74 and 86.
Solution: Factors of 74: 1, 2, 37, 74. Factors of 86: 1, 2, 43, 86. Common factors: 1 and 2.
Applications: HCF, LCM, and Real-Life Use
Knowing the factors of 74 is useful when finding the HCF (Highest Common Factor) or LCM (Lowest Common Multiple) with other numbers. For example, to find HCF of 74 and 100:
1. List factors of both numbers: 74 (1, 2, 37, 74) and 100 (1, 2, 4, 5, 10, 20, 25, 50, 100).2. The highest common factor is 2.
You may also use factors for organizing objects equally, simplifying fractions, or dividing items in real life.
Frequent Errors and Misunderstandings
- Missing a factor pair (e.g., not listing 2 and 37).
- Confusing multiples with factors. (Remember: Factors are numbers that divide; multiples are the result of multiplying 74 with integers.)
- Assuming prime factorization is the same as listing all factors.
Try These Yourself
- Write all the negative factors of 74.
- Verify if 148 is a multiple or factor of 74.
- Find the HCF of 74 and 92.
- Check if 37 is a factor of 74 (show steps).
- List the next three multiples of 74 beyond 370.
Related Concepts and Further Learning
Learning about factors of 74 helps you with other maths chapters. If you want to practice further:
- Explore Factors of 72 and Factors of 75 to compare nearby numbers
- Read about Prime Factors to strengthen your understanding of breaking numbers down
- Use HCF of Two Numbers for more practice questions
- Practice building your own Factor Tree for different numbers
Vedantu’s maths educators often use visual cues and step-by-step tricks to help you master factorization quickly, especially for numbers like 74.
We explored factors of 74—from definition, calculation, factor pairs, negative factors, prime factorization, and their applications. Continue practicing with Vedantu to boost your calculation speed and ace your exams!
FAQs on Factors of 74: How to Find, Pair, and Use Them in Maths
1. What are the factors of 74?
The factors of 74 are the whole numbers that divide 74 without leaving a remainder. These are 1, 2, 37, and 74. Factors are also known as divisors.
2. How do I find the factors of 74 using the division method?
To find the factors using division, systematically divide 74 by each whole number, starting from 1, until you reach 74. If the division results in a whole number quotient with no remainder, then the divisor is a factor. For 74, the factors are found by dividing by 1, 2, 37, and 74.
3. What are the factor pairs of 74?
Factor pairs are sets of two numbers that multiply to give the original number. The factor pairs of 74 are (1, 74) and (2, 37). Remember that negative pairs (-1, -74) and (-2, -37) also exist.
4. What is the prime factorization of 74?
Prime factorization expresses a number as a product of its prime factors (numbers divisible only by 1 and themselves). The prime factorization of 74 is 2 x 37.
5. Is 74 a prime or composite number?
74 is a composite number because it has more than two factors (1, 2, 37, and 74).
6. How are the factors of 74 used to find the Highest Common Factor (HCF)?
To find the HCF of 74 and another number, list the factors of both numbers. The largest factor common to both lists is the HCF. For example, to find the HCF of 74 and 92, the factors of 74 are 1, 2, 37, 74, and the factors of 92 are 1, 2, 4, 23, 46, 92. The highest common factor is 2.
7. How are the factors of 74 used to find the Lowest Common Multiple (LCM)?
The LCM of 74 and another number can be found using the prime factorization of both numbers. First find the prime factorization of both numbers, and then multiply the highest power of each prime factor present in either factorization. For example, to find the LCM of 74 and 12, the prime factorization of 74 is 2 x 37, and the prime factorization of 12 is 22 x 3. Therefore, the LCM is 22 x 3 x 37 = 444.
8. What are the first five multiples of 74?
Multiples of 74 are numbers obtained by multiplying 74 by whole numbers. The first five multiples of 74 are: 74, 148, 222, 296, and 370.
9. What is the sum of all the factors of 74?
To find the sum of all factors of 74, add all its factors together: 1 + 2 + 37 + 74 = 114
10. Explain how to use a factor tree to find the prime factors of 74.
A factor tree visually represents the prime factorization. Start with 74. Since 74 is even, it's divisible by 2. 74 = 2 x 37. Both 2 and 37 are prime numbers, so the prime factorization is complete. The prime factors are 2 and 37.
11. Are all factors of 74 even numbers?
No. While 2 and 74 are even factors, 1 and 37 are odd factors of 74.
