How to Find All Factors of 67 Step by Step?
The concept of factors of 67 is essential in mathematics and helps in solving real-world and exam-level problems efficiently.
Understanding Factors of 67
A factor of 67 refers to any whole number that divides 67 exactly, without leaving a remainder. This concept is widely used in division, prime number identification, and the number system. In mathematics, finding all the factors of a number like 67 helps students understand divisibility, primes, and composite numbers.
What are the Factors of 67?
The factors of 67 are the numbers that divide 67 with zero remainder. Since 67 is a prime number, its only factors are 1 and 67 itself.
List of Factors of 67
Here is the complete list of the factors of 67:
2. 67
No other positive number divides 67 exactly.
Is 67 a Prime or Composite Number?
67 is a prime number because it has only two distinct factors: 1 and itself. In mathematics, a composite number has more than two factors, whereas a prime has exactly two.
| Number | Prime? | Composite? | Factors |
|---|---|---|---|
| 67 | Yes | No | 1, 67 |
| 12 | No | Yes | 1, 2, 3, 4, 6, 12 |
How to Find the Factors of 67
To find the factors of 67 step by step, use the division method as follows:
2. Try 2.
3. Check all numbers up to √67 ≈ 8.1, i.e., test 3, 4, 5, 6, 7, 8.
4. Lastly, try 67 itself.
Therefore, the factors of 67 are just 1 and 67.
Prime Factorization of 67
Since 67 is a prime number, its prime factorization is simply:
There is no further factor tree for 67 because it cannot be split into smaller prime numbers.
Factor Pairs of 67
A factor pair is a set of two numbers whose product is 67.
| Positive Pairs | Negative Pairs |
|---|---|
| 1 × 67 | -1 × -67 |
| 67 × 1 | -67 × -1 |
These are the only pairs for 67 since it is prime.
Worked Example – Checking Factors
Find all factors of 67 and explain why no other numbers are factors.
2. Test 2: 67 ÷ 2 = 33.5. (Not a factor)
3. Test 3: 67 ÷ 3 ≈ 22.33. (Not a factor)
4. Test numbers up to 8 — none divide 67.
5. Test 67: 67 ÷ 67 = 1. (Factor)
Final answer: Only 1 and 67 exactly divide 67, so these are its only factors.
Sum and Properties of Factors of 67
Sum of factors of 67 = 1 + 67 = 68.
The greatest common factor (GCF) of 67 with a different prime like 71 is 1.
If another number shares 1 and 67 as factors (e.g., 201 = 67 × 3), then 67 is also a common factor.
Comparison: Neighbor Numbers and Their Factors
Comparing the factors of 67 with nearby numbers helps in recognizing patterns:
| Number | Factors | Prime/Composite |
|---|---|---|
| 66 | 1, 2, 3, 6, 11, 22, 33, 66 | Composite |
| 67 | 1, 67 | Prime |
| 68 | 1, 2, 4, 17, 34, 68 | Composite |
Quick Practice Questions
2. Is 67 a composite number?
3. What is the sum of all factors of 67?
4. Which pairs multiply together to form 67?
5. Does 67 have any even factors?
Common Mistakes to Avoid
- Thinking 67 has more than two factors because it is a “large” number.
- Confusing factors with multiples (for example: 134 is a multiple, not a factor).
- Assuming 67 can be divided by 13, 11, or 17 without checking division properly.
Real-World Applications
The concept of factors of numbers like 67 appears in areas such as cryptography, data grouping, and creating number puzzles. It is also useful in simplifying math tasks for competitive exams. Vedantu helps students see these patterns and master the number system for exams and practical tasks.
We explored the idea of factors of 67, how to find them, their properties, and why 67 is prime. Practicing questions and comparing numbers strengthens these concepts, which are important for school exams and real-life math use. Practice regularly with Vedantu for a strong foundation in factors and prime numbers.
Related Maths Resources on Vedantu
- Prime Numbers
- Factors of a Number
- Factors of 68
- Factors of 12
- Common Factors
- Even and Odd Numbers
- Multiples of 4
- Factors of 60
- Prime and Composite Numbers
- Factors of 23
FAQs on Factors of 67: Complete Guide, Prime Check, List & Methods
1. What are the factors of 67?
The factors of 67 are the numbers that divide 67 exactly without leaving any remainder. Since 67 is a prime number, it has only two factors: 1 and 67.
2. Is 67 a prime or composite number?
67 is a prime number because it has exactly two factors: 1 and itself (67). It is not divisible by any other natural number.
3. How can you find the factors of 67?
To find the factors of 67, use the division method by dividing 67 by natural numbers starting from 1 up to 67. Only numbers that divide 67 without a remainder are factors. Since only 1 and 67 satisfy this, these are the factors of 67.
4. What are the factor pairs of 67?
The factor pairs of 67 are pairs of numbers that multiply to 67. Because 67 is prime, it has only one pair: (1, 67). The negative factor pairs are (-1, -67).
5. In which multiplication table does 67 appear?
67 appears in its own multiplication table (Table of 67). It is divisible only by 1 and 67, so it does not appear in other multiplication tables.
6. What is the prime factorization of 67?
Since 67 is a prime number, its prime factorization is simply 67 itself. It cannot be broken down further into other prime factors.
7. Why does 67 have only two factors?
67 has only two factors because it is a prime number. By definition, prime numbers have exactly two factors: 1 and the number itself, meaning they are divisible only by those two numbers.
8. Why can't 67 form a factor tree with branches?
A factor tree branches when a number is composite and can be broken down into smaller factors. Since 67 is prime, it cannot be factored into smaller natural numbers, so its factor tree consists of only one node: 67 itself.
9. Why do students confuse factors and multiples of 67?
Students often confuse factors and multiples because both involve division and multiplication concepts. Factors divide a number exactly (like 1 and 67 for 67), while multiples are products of that number with other natural numbers (e.g., 67, 134, 201).
10. Why is 67 sometimes mistaken for a composite number?
67 may be mistaken as composite because it is a large odd number, unlike smaller primes most students first learn. Without checking divisibility properly, students might assume it has more factors, but detailed divisibility tests confirm its prime status.
11. Why is memorizing prime numbers important for exams?
Memorizing prime numbers helps in quick factorization, simplifying fractions, and solving divisibility problems in exams. Knowing key primes like 67 saves time and improves accuracy during board and competitive tests.
12. What common errors happen in identifying factors of numbers like 67?
Common errors include: assuming all odd numbers are composite, confusing factors with multiples, and neglecting to check divisibility properly. To avoid mistakes, always perform step-by-step division tests and remember the definition of prime numbers.
