What are the factors and factor pairs of 91?
The concept of factors of 91 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Quickly finding factors is important for homework, MCQ preparation, and understanding divisibility, prime factorization, and LCM/HCF related topics.
What Is Factors of 91?
A factor of 91 is a natural number that divides 91 exactly with no remainder. You’ll find this concept applied in areas such as prime factorization, divisibility tests, and finding common factors in LCM and HCF problems. Knowing factors is also helpful in mental maths and competitive exam questions.
Key Formula for Factors of 91
Here’s the standard formula: If \( a \times b = 91 \), then a and b are pair factors of 91. To find all factors, list every number that divides 91 without leaving a remainder.
All Factors of 91 (Quick Answer)
- List of factors of 91: 1, 7, 13, 91
- Prime factors of 91: 7, 13
- Is 91 a prime number? No (it is composite)
- Factor pairs of 91: (1, 91) and (7, 13)
| Factor | Pair |
|---|---|
| 1 | 91 |
| 7 | 13 |
Step-by-Step Illustration
- List numbers from 1 up to 91.
Start with 1: 91 ÷ 1 = 91 (factor pair 1 and 91) - Check small prime numbers:
2: 91 ÷ 2 = 45.5 (not an integer, so not a factor)3: 91 ÷ 3 ≈ 30.33 (not a factor)5: 91 ÷ 5 = 18.2 (not a factor)7: 91 ÷ 7 = 13 (both integers, factor pair 7 and 13) - Continue up to √91 (about 9.5), as higher numbers will repeat pairs.
Nothing else up to 9 is an exact factor. List all pairs you found: 1, 7, 13, 91.
Prime Factorization of 91
The prime factorization of 91 breaks it into only prime numbers multiplied together:
1. Start: 91 ÷ 7 = 13 (since 7 is a prime factor)
2. 13 is a prime number and cannot be divided further (except by 1 and itself).
So, 91 = 7 × 13. The prime factors of 91 are 7 and 13.
Divisibility Properties
| Number | Is a Factor of 91? | Reason |
|---|---|---|
| 1 | Yes | Every number is divisible by 1 |
| 3 | No | 91 ÷ 3 = 30.33 (not whole) |
| 7 | Yes | 91 ÷ 7 = 13 |
| 13 | Yes | 91 ÷ 13 = 7 |
| 17 | No | 91 ÷ 17 = 5.35 (not whole) |
| 91 | Yes | A number divides itself exactly |
Speed Tricks For Factorization
To quickly check divisibility of 91 by 7, double the unit digit (1×2=2), subtract from the rest (9-2=7)—because 7 divides 7, 91 is divisible by 7. This shortcut saves time in MCQs!
Related Factor & LCM/HCF Problems
- Find HCF of 91 and 49
Factors of 91: 1, 7, 13, 91; Factors of 49: 1, 7, 49. HCF is 7. - What is the LCM of 91 and 13?
LCM(91, 13) = (91 × 13)/HCF(91,13) = (91×13)/13 = 91
Try These Yourself
- List all factors of 91 and 81—find common ones
- Check if 14 is a factor of 91
- Write the prime factorization of 91 using a factor tree
- How many factor pairs does 91 have?
Frequent Errors and Misunderstandings
- Thinking 91 is a prime number—because it is not even or a multiple of 5 (always check divisibility up to √n!)
- Missing the factors 7 and 13 due to poor division skills
Relation to Other Concepts
The idea of factors of 91 connects closely with prime numbers (since 91’s factors are both primes), HCF/LCM problems, and factors and multiples. Mastering factors helps with number patterns, divisibility tests, and more advanced Maths chapters.
Classroom Tip
Remember, a composite number like 91 can be checked quickly for prime factors by dividing by small primes up to its square root. Vedantu’s teachers recommend using pair factor tables and factor trees in your notes for fast revision!
We explored factors of 91—from definition, formula, quick answer list, pair tables, tricks, and related LCM/HCF uses, plus common mistakes and tips to remember fast. For deeper conceptual understanding and more practice, check out Vedantu’s live doubt-solving sessions and Maths pages.
Frequently Asked Questions About Factors of 91
FAQs
Q1: What are all the factors of 91?
A1: The factors of 91 are 1, 7, 13, and 91.
Q2: Is 91 a prime number?
A2: No, 91 is not a prime number; it is composite because it has factors besides 1 and itself.
Q3: What is the prime factorization of 91?
A3: 91 can be written as 7 × 13, where both numbers are prime.
Q4: How do I quickly check if 7 is a factor?
A4: Yes, because 91 ÷ 7 = 13 (exact and no remainder).
Q5: What are factor pairs of 91?
A5: The pairs are (1, 91) and (7, 13).
Vedantu Maths Topic Interlinks
- Factors of 81 – Compare square factorization with 91.
- Prime Numbers – Understand how 7 and 13 are primes.
- HCF and LCM of Two Numbers – Find common factors and LCM/HCF using 91.
- Prime Factorization – Step-by-step process for any composite number.
- Factors and Multiples – Clear up differences and deepen number sense.
FAQs on Factors of 91: Definition, Prime Factorization & Easy Methods
1. What are the factors of 91?
The factors of 91 are 1, 7, 13, and 91. These are all the whole numbers that divide 91 without leaving a remainder.
2. What is the prime factorization of 91?
The prime factorization of 91 is 7 × 13. This means that 7 and 13 are the only prime numbers that multiply together to give 91.
3. Is 91 a prime number?
No, 91 is a composite number. A prime number has only two factors: 1 and itself. Since 91 has more than two factors (1, 7, 13, and 91), it's composite.
4. What are the pairs of factors of 91?
The pairs of factors of 91 are (1, 91) and (7, 13). When you multiply the numbers in each pair, you get 91.
5. How do I find the factors of 91?
To find the factors of 91, systematically divide 91 by each whole number, starting from 1. If the division results in a whole number (no remainder), that number is a factor. Continue this process until you reach the square root of 91 (approximately 9.5).
6. Is 91 divisible by 3?
No, 91 is not divisible by 3. A quick divisibility rule for 3 is to check if the sum of the digits is divisible by 3. In 91, 9 + 1 = 10, which is not divisible by 3.
7. Is 91 divisible by 7?
Yes, 91 is divisible by 7. 91 ÷ 7 = 13.
8. Is 17 a factor of 91?
No, 17 is not a factor of 91. 91 divided by 17 leaves a remainder.
9. How is knowing the factors of 91 useful in mathematics?
Understanding the factors of 91 is crucial for various mathematical operations, including finding the **Highest Common Factor (HCF)** and **Lowest Common Multiple (LCM)** of numbers, simplifying fractions, and solving problems related to **divisibility** and **prime factorization**.
10. What is the difference between factors and multiples of 91?
Factors of 91 are numbers that divide 91 exactly (with no remainder), such as 1, 7, 13, and 91. Multiples of 91 are numbers obtained by multiplying 91 by whole numbers (e.g., 91, 182, 273, etc.).
11. Can you explain the divisibility rule for 7 in relation to 91?
One method: Double the last digit (1 x 2 = 2) and subtract it from the remaining digits (9 - 2 = 7). If the result (7 in this case) is divisible by 7, then the original number (91) is also divisible by 7. Another method involves repeatedly subtracting twice the last digit from the remaining digits until a small number divisible by 7 is obtained.
12. How are the factors of 91 related to its prime factors?
The prime factors of 91 (7 and 13) are the fundamental building blocks from which all other factors are derived. Every factor of 91 can be expressed as a product of these prime factors or 1.
