What Are the Factors of 67 and Why Is 67 a Prime Number
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 Explained with Prime Insight
1. What are the factors of 67?
The factors of 67 are 1 and 67 only. Since 67 is a prime number, it is divisible only by 1 and itself. Therefore, the complete list of positive factors is:
- 1
- 67
2. Is 67 a prime number?
Yes, 67 is a prime number because it has exactly two distinct positive factors: 1 and 67. A prime number is defined as a number greater than 1 that is divisible only by 1 and itself. Since no other number divides 67 evenly, it is prime.
3. Why does 67 have only two factors?
The number 67 has only two factors because it is a prime number. Prime numbers are divisible only by:
- 1
- The number itself
4. How do you find the factors of 67?
To find the factors of 67, check which numbers divide 67 exactly without leaving a remainder.
- Divide 67 by 1 → 67 ÷ 1 = 67 (no remainder)
- Test other numbers like 2, 3, 4, 5, etc.
- None divide evenly except 67 itself
5. What is the prime factorization of 67?
The prime factorization of 67 is simply 67. Since 67 is already a prime number, it cannot be broken down further into smaller prime factors. In exponential form, it is written as:
- 67 = 67¹
6. What are the factor pairs of 67?
The only factor pair of 67 is (1, 67). A factor pair consists of two numbers that multiply together to give the original number. Since 67 is prime, no other pair of whole numbers multiplies to 67.
7. Does 67 have any negative factors?
Yes, the negative factors of 67 are -1 and -67. Every positive factor has a corresponding negative factor because:
- (-1) × (-67) = 67
8. Is 67 divisible by 3, 5, or 7?
No, 67 is not divisible by 3, 5, or 7. Checking divisibility:
- 67 ÷ 3 leaves a remainder
- 67 does not end in 0 or 5, so not divisible by 5
- 67 ÷ 7 leaves a remainder
9. What is the sum of the factors of 67?
The sum of the factors of 67 is 68. Since the only positive factors are 1 and 67, we add them:
- 1 + 67 = 68
10. How many total factors does 67 have?
The number 67 has exactly 2 positive factors. These are:
- 1
- 67
