How to Find Factors of 71 Step by Step with Examples
The concept of factors of 71 is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Knowing the factors of a number like 71 can make arithmetic, number theory, and divisibility questions much simpler to approach, both in school and competitive exams.
Understanding Factors of 71
A factor of 71 is any whole number that divides 71 exactly without leaving a remainder. This concept is widely used in prime factorization, divisibility rules, and finding highest common factors (HCF). Since 71 is a special kind of number, let’s look closer at how its factors are determined.
How to Find Factors of 71 – Stepwise Method
To find the factors of 71, use the division method:
1. Start by dividing 71 by 1: 71 ÷ 1 = 71, remainder 0.
2. Next, check all whole numbers up to 71 (2, 3, 4, ..., 71).
3. For each number, see if it divides 71 evenly with no remainder.
4. If it does, it’s a factor; if not, ignore it.
5. 71 is not divisible by 2 (gives 35.5), by 3 (gives decimal), by 4 (gives decimal), and so on until 71 itself.
6. 71 ÷ 71 = 1, remainder 0.
7. Therefore, the only factors of 71 are 1 and 71.
Because it has only two factors, 71 is called a prime number.
List of Factors and Factor Pairs of 71
Here’s a helpful table to understand the factors of 71 more clearly:
Factors of 71 and Their Pairings
| Factor | Pair Factor | Product (Factor × Pair Factor) |
|---|---|---|
| 1 | 71 | 71 |
| 71 | 1 | 71 |
| -1 | -71 | 71 |
| -71 | -1 | 71 |
As shown, the only factor pairs of 71 are (1, 71) and (71, 1). Their negative pairs (-1, -71) and (-71, -1) are also valid in advanced contexts.
Prime Factorization of 71
The prime factorization of 71 is especially simple:
1. Start with 71.
2. See if it divides by 2 – no.
3. Check next highest primes (3, 5, 7, 11, 13, 17, etc.) – none divide exactly.
4. Therefore, 71 is only divisible by itself and 1.
5. So, the prime factorization is only 71:
Therefore, 71 itself is a prime factor and does not break down further. For more on how prime numbers are identified, see Prime Numbers.
Comparison: Factors of 70, 71, and 72
Let’s compare factors of 71 to those of nearby numbers:
| Number | Factors List | Prime? |
|---|---|---|
| 70 | 1, 2, 5, 7, 10, 14, 35, 70 | No |
| 71 | 1, 71 | Yes |
| 72 | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 | No |
As you can see, 71 is unique compared to its neighbors. For more, compare with Factors of 70 and Factors of 72.
Worked Example – Solving Problems with Factors of 71
Let’s solve an example step by step:
Example: Find all factors of 71.
1. List all numbers from 1 to 71.
2. Test each number to see if it divides 71 evenly (i.e., 71 ÷ number has no remainder).
3. 1 divides 71 (71 ÷ 1 = 71) → Factor: 1
4. Numbers 2 to 70 do not divide 71 evenly (71 ÷ any of 2–70 gives decimals).
5. 71 divides itself (71 ÷ 71 = 1) → Factor: 71
6. List: 1, 71.
Final answer: The factors of 71 are 1 and 71.
Practice Problems
- Is 71 a composite number or a prime number? Explain why.
- List all factors of 81 and compare them with 71. Which has more?
- Find the HCF of 71 and 17.
- What is the sum of all factors of 71?
- Are there any numbers between 1 and 71, apart from 1 and 71, that divide 71 completely?
Common Mistakes to Avoid
- Thinking 71 is composite because it is a two-digit number.
- Assuming every number between 1 and itself is a factor for all numbers.
- Confusing factors with multiples (multiples of 71 are not the same as its factors).
Real-World Applications
The concept of factors of 71 (and of prime numbers in general) is found in cryptography, coding, computer science, and even scheduling tournaments or distributing resources evenly. At Vedantu, students can learn how such maths ideas help in both academic and practical scenarios. For a deeper look into related concepts, visit Factors of a Number and Factors and Multiples.
We explored the idea of factors of 71, how to apply it, solve related problems, and understand its real-life relevance. Practice more with Vedantu to build confidence in these concepts. For further revision, compare with Factors of 81, Factors of 17, and Factors of 91 to build a strong foundation in factors and related arithmetic topics.
FAQs on What Are the Factors of 71?
1. What are the factors of 71?
The factors of 71 are 1 and 71 itself. Since 71 is a prime number, it has only these two distinct factors that divide it exactly without leaving a remainder.
2. Is 71 a prime number?
Yes, 71 is a prime number because it has exactly two factors: 1 and 71. It cannot be divided evenly by any other whole number.
3. How to find the factors of 71 by division?
To find the factors of 71 using the division method, divide 71 by integers starting from 1 up to 71. If the division leaves no remainder, the number is a factor. For 71, only 1 and 71 divide it exactly, confirming it is prime.
4. What are the factor pairs of 71?
The factor pairs of 71 are the pairs of numbers that multiply to give 71. Since 71 is prime, it has only one factor pair: (1, 71). Considering negative factors, the pairs are (-1, -71) as well.
5. What is the HCF of 71 and other numbers?
The Highest Common Factor (HCF) of 71 with any other number is found by listing the factors of both numbers and choosing the greatest one they share. Since 71 is prime, the HCF with any number not divisible by 71 is always 1.
6. Is 71 a multiple of 8?
No, 71 is not a multiple of 8 because when 71 is divided by 8, it leaves a remainder. Multiples of 8 are numbers exactly divisible by 8, like 8, 16, 24, and so on.
7. Why is 71 not considered a composite number?
A number is composite if it has more than two factors. Since 71 has only two factors—1 and 71—it does not meet the criteria and is therefore a prime, not composite, number.
8. Why do students sometimes confuse factors with multiples?
Students may confuse factors with multiples because both involve division and multiplication. Factors divide a number exactly, whereas multiples are numbers obtained by multiplying that number with integers. Clear conceptual distinction helps avoid this confusion.
9. Can 71 be factorized by any number apart from 1 and itself?
No, 71 cannot be factorized by any other number except 1 and 71. This is because it is prime and has no other divisors.
10. Are there real-world applications for recognizing the factors of 71?
Yes, recognizing the factors of 71 is useful in areas like cryptography, number theory, and problem solving where identifying prime numbers plays a critical role in encryption algorithms, coding, and optimization tasks.
11. How to quickly revise prime and composite numbers for exams?
To quickly revise prime and composite numbers for exams:
1. Memorize the list of prime numbers up to 100.
2. Practice divisibility rules for common numbers.
3. Use factor trees to break down composite numbers.
4. Solve sample problems to reinforce concepts.
5. Use flashcards or apps for quick revision.
