How to Find All Factors and Factor Pairs of 75 (With Examples)
The concept of factors of 75 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you are preparing for school tests, Olympiads, or seeking quick problem-solving methods, understanding the factors of 75 can greatly improve your mathematical confidence.
What Is Factors of 75?
A factor of 75 is any whole number that divides 75 exactly, leaving no remainder. In other words, if you multiply two numbers and get 75 as the answer, both are called factors of 75. You’ll find this concept applied in areas such as multiples and divisibility, prime factorization, and finding LCM or HCF in arithmetic and algebra.
Key Formula for Factors of 75
Here’s the standard way to represent the factors of 75:
\( \text{If}~ 75 \div n = \text{whole number},~ \text{then}~ n~ \text{is a factor of}~ 75 \)
Why Are Factors of 75 Important?
Knowing the factors of 75 helps in simplifying fractions, solving algebraic problems, and even distributing objects evenly. You’ll need to use factors often in both school exams and competitive tests. Whenever you work with time, money, or measurements that can be split into equal parts, the idea of factors comes into play.
How to Find Factors of 75: Step-by-Step
- Start with 1: 75 ÷ 1 = 75. So, 1 and 75 are factors.
- Try 2: 75 ÷ 2 = 37.5 (not a whole number)
- Try 3: 75 ÷ 3 = 25. So, 3 and 25 are also factors.
- Try 4: 75 ÷ 4 = 18.75 (not a whole number)
- Try 5: 75 ÷ 5 = 15. So, 5 and 15 are factors.
- Go further, but once you reach numbers greater than 15, repeated factors start. List them all:
So, the complete factors of 75 are: 1, 3, 5, 15, 25, 75.
Prime Factorization of 75
To break 75 into its prime factors:
- Start by dividing by the smallest prime:
75 ÷ 3 = 25 - Continue with next prime factors:
25 ÷ 5 = 55 ÷ 5 = 1
So, the prime factorization of 75 is:
3 × 5 × 5 or \( 3 \times 5^2 \)
Factor Pairs of 75
| Pair | Result |
|---|---|
| 1 × 75 | 75 |
| 3 × 25 | 75 |
| 5 × 15 | 75 |
Speed Trick or Vedic Shortcut
To check if a number is a factor quickly, see if it divides 75 with no remainder. For larger numbers, use the divisibility rule: If a number ends with 5 or 0, it’s always divisible by 5. If the sum of digits is a multiple of 3, it’s divisible by 3 as well. Tricks like these are popular for quick checks in competitive exams. Vedantu’s live lessons often include such simple speed tricks!
Solved Examples
- Check if 15 is a factor of 75.
75 ÷ 15 = 5. Yes, 15 is a factor. - Find all even factors of 75.
75 ÷ 2 = 37.5, 75 ÷ 4 = 18.75, etc. No even factors besides 75 itself, which is odd. - List the prime factors of 75.
Prime factors: 3, 5, 5.
Try These Yourself
- Find the sum of all factors of 75.
- Which are the factors of 75 between 10 and 30?
- Is 25 a factor of 75? Prove your answer.
- List all factors of 75 and 90. Which are common?
Frequent Errors and Misunderstandings
- Missing pairs by forgetting to check numbers beyond 10.
- Assuming every divisor is a prime or getting confused between factors and multiples.
- Mixing up factor pairs (e.g., 5 × 15 and 15 × 5 are the same pair).
Relation to Other Concepts
The idea of factors of 75 fits into broader number operations. It is connected with divisibility, multiples, greatest common factor (HCF), least common multiple (LCM), and prime decomposition. Mastering this helps students confidently approach topics like prime factorization and factors and multiples in later classes.
Classroom Tip
A quick way to remember factors: Every factor comes in a pair. Start with 1 and go upwards. When multiplication repeats (like 15 × 5 and 5 × 15), you’ve found all pairs. Vedantu’s expert teachers often use factor trees and quick worksheets to help you learn this fast in class!
We explored factors of 75—from definition, formula, examples, and mistakes, to quick connections. Continue practicing with Vedantu to become confident in finding factors and using them in all maths problems!
Related Maths Topics
FAQs on Factors of 75: Definition, Steps & Prime Factorization
1. What are the factors of 75?
The factors of 75 are the whole numbers that divide 75 without leaving a remainder. These are: 1, 3, 5, 15, 25, and 75.
2. How do you find all the factors of 75?
To find all factors, systematically check for divisibility. Start with 1 and proceed, checking if each number divides 75 evenly. Alternatively, find pairs of numbers that multiply to 75. You can also use a factor tree to break 75 down into its prime factors (3 x 5 x 5), then combine those factors in different ways to find all the factors.
3. What is the prime factorization of 75?
The prime factorization of 75 is 3 x 5 x 5 or 3 x 52. This means 75 can be expressed as the product of only prime numbers.
4. What are the factor pairs of 75?
The factor pairs of 75 are pairs of numbers that multiply to 75. These are: (1, 75), (3, 25), and (5, 15). Remember that (15,5), (25,3), and (75,1) are considered the same pairs.
5. Is 75 a prime number?
No, 75 is not a prime number. A prime number has only two factors: 1 and itself. Since 75 has more than two factors (1, 3, 5, 15, 25, and 75), it's a composite number.
6. How are factors of 75 useful in finding the Highest Common Factor (HCF) or Lowest Common Multiple (LCM)?
Finding the factors of 75 (and other numbers) is crucial for determining the HCF and LCM. The HCF is the largest factor common to two or more numbers. The LCM is the smallest multiple common to two or more numbers. By comparing the factors of 75 with the factors of other numbers, you can identify the HCF and build towards calculating the LCM.
7. What is the difference between a factor and a multiple?
A **factor** of a number divides that number exactly, leaving no remainder. A **multiple** of a number is obtained by multiplying that number by any whole number. For example, 3 and 25 are factors of 75, while 150 and 225 are multiples of 75.
8. What are some real-world applications of understanding factors?
Understanding factors is useful in many areas, such as dividing quantities equally (e.g., sharing 75 sweets among friends), simplifying fractions, solving algebraic equations, and understanding ratios and proportions.
9. Can you explain how to use a factor tree to find the factors of 75?
A factor tree visually breaks down a number into its prime factors. For 75:
- Start with 75.
- Find a pair of factors (e.g., 3 and 25).
- If a factor is not prime (like 25), break it down further (5 and 5).
- Continue until all branches end with prime numbers (3, 5, 5).
- Then, combine these prime factors (3 x 5 x 5) and their various combinations to list all factors of 75 (1, 3, 5, 15, 25, 75).
10. What are the odd factors of 75?
All the factors of 75 (1, 3, 5, 15, 25, 75) are odd numbers because 75 itself is an odd number. An even number will always have at least one even factor (2).
11. Explain how to find the factors of a number using division.
To find the factors of a number using division, divide the number by each whole number, starting from 1, up to the number itself. If the division results in a whole number (no remainder), then that number is a factor. For 75: Divide 75 by 1, 2, 3... until you reach 75. The numbers that divide evenly are the factors.
