How to Find the Factors of 68 Step by Step
The concept of factors of 68 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Learning to find factors quickly also helps in topics like HCF, LCM, and simplifying fractions.
What Are Factors of 68?
A factor of 68 is a whole number that divides 68 exactly, leaving no remainder. These factors include all positive integers that can multiply in pairs to make 68 (such as 2 × 34). Understanding the factors of 68 is helpful when working with other numbers' factors, doing HCF/LCM calculations, and checking divisibility in maths problems.
All Factors of 68
The factors of 68 are the numbers that divide it evenly. Here’s the complete list of positive factors:
- 1
- 2
- 4
- 17
- 34
- 68
So, 68 has exactly 6 positive factors. Each of these divides 68 with zero remainder. Negative factors also exist (e.g. -2), but in most schoolwork, we use positive ones.
Factors of 68 in Pairs
Some students find it easier to remember factors as pairs of numbers whose product is 68.
| Factor 1 | Factor 2 | Multiplication |
|---|---|---|
| 1 | 68 | 1 × 68 = 68 |
| 2 | 34 | 2 × 34 = 68 |
| 4 | 17 | 4 × 17 = 68 |
This method is especially handy for visual learners and those revising on mobile devices.
Prime Factorization of 68
Prime factorization breaks 68 into a product of its prime factors. Here’s how you can do it step by step:
1. Divide 68 by the smallest prime number (2): 68 ÷ 2 = 342. Divide 34 by 2 again: 34 ÷ 2 = 17
3. 17 is already a prime number.
So, the prime factorization of 68 is: 2 × 2 × 17 or 2² × 17.
Prime factors of 68 are 2 and 17. This break-down is very helpful for advanced maths and quick division tricks.
Is 68 a Prime or Composite Number?
68 is a composite number because it has more than two factors (not just 1 and itself). Prime numbers have exactly two factors. Since 68 has 6 factors, it is definitely composite.
Is 68 a Perfect Square?
No, 68 is not a perfect square. The square root of 68 is approximately \(\sqrt{68} ≈ 8.246\), which is not an integer. (8 × 8 = 64 and 9 × 9 = 81, so 68 is between those.)
Finding Factors of 68 by Division Method
You can also use the division method to check each number from 1 up to 68:
1. Start with 1: 68 ÷ 1 = 68 (no remainder, so 1 is a factor)2. Try 2: 68 ÷ 2 = 34 (no remainder, so 2 is a factor)
3. Try 3: 68 ÷ 3 ≈ 22.67 (not a whole number, so 3 is NOT a factor)
4. Continue this way up to 68. The whole-number results (no remainders) are all factors: 1, 2, 4, 17, 34, 68.
This approach works for any natural number!
Applications of Factors of 68
Knowing the factors of 68 is useful for:
- Calculating HCF and LCM with other numbers
- Simplifying fractions (like 68/34 = 2)
- Finding divisibility patterns
- Solving questions in exams and Olympiads
You’ll use these skills in many chapters—especially when working with word problems!
Solved Examples: Factors of 68
Example 1: Find the sum of all positive factors of 68.
1. List out the factors: 1, 2, 4, 17, 34, 68
2. Add them: 1 + 2 + 4 + 17 + 34 + 68 = 126
Example 2: What is the greatest common factor (GCF) of 68 and 34?
1. Factors of 68: 1, 2, 4, 17, 34, 68
2. Factors of 34: 1, 2, 17, 34
3. Common factors: 1, 2, 17, 34
4. Greatest one: 34
Try These Yourself
- List all even factors of 68.
- Check if 7 is a factor of 68.
- Find all common factors of 68 and 60.
- Write the prime factorization of 68 in words.
- Pair up all factors of 68 in negative form.
Frequent Errors and Misunderstandings
- Mixing up multiples and factors (e.g. thinking 136 is a factor—actually it’s a multiple).
- Missing factors (like forgetting 17 or 34).
- Assuming perfect square status (root is not an integer).
Relation to Other Concepts
The idea of factors of 68 is closely linked to Factors of 34, prime factorization, even/odd numbers, and divisibility rules. Knowing how to break down numbers makes all of maths easier!
Classroom Tip
To quickly list the factors of any even number, start with 1 and itself, then try dividing by 2, then other prime numbers like 4 and 17. Repeat with all numbers up to the square root of 68. Vedantu’s teachers often use visual factor trees and tables for this in live classes.
We explored factors of 68—from the definition to factor pairs and prime factorization, with handy tips and common mistakes. Keep learning with Vedantu to build confidence in Maths and be ready for any school or competitive exam!
Related learning for deeper understanding:
FAQs on What Are the Factors of 68?
1. What are the factors of 68?
The factors of 68 are the whole numbers that divide 68 without leaving a remainder. These are: 1, 2, 4, 17, 34, and 68. Note that 68 also has negative factors: -1, -2, -4, -17, -34, and -68.
2. What are the factor pairs of 68?
Factor pairs of 68 are pairs of numbers that multiply to 68. The positive factor pairs are: (1, 68), (2, 34), and (4, 17). Negative factor pairs also exist, such as (-1, -68), (-2, -34), and (-4, -17).
3. What is the prime factorization of 68?
The prime factorization of 68 expresses it as a product of its prime factors. It is 2 x 2 x 17, or 2² x 17. This means 68 is only divisible by the prime numbers 2 and 17.
4. Is 68 a perfect square?
No, 68 is not a perfect square. A perfect square is a number that can be obtained by squaring a whole number. The square root of 68 is approximately 8.246, which is not a whole number.
5. What numbers is 68 divisible by?
68 is divisible by its factors: 1, 2, 4, 17, 34, and 68. It's also divisible by the negative versions of these numbers.
6. Is 68 a prime or composite number?
68 is a composite number. A composite number has more than two factors (1 and itself). Since 68 has several factors (1, 2, 4, 17, 34, 68), it's composite.
7. How do I find all the factors of 68 using the division method?
To find all factors using division: Divide 68 by each whole number starting from 1, and record the quotients that are also whole numbers. These quotients are the factors. For 68: 68 ÷ 1 = 68; 68 ÷ 2 = 34; 68 ÷ 4 = 17; You'll find the factors 1, 2, 4, 17, 34, and 68.
8. How can I use a factor tree to find the prime factorization of 68?
Start by breaking down 68 into two factors. Since 68 is even, use 2: 68 = 2 x 34. Then break down 34 (also even) into 2 x 17. Since 17 is a prime number, the factorization is complete: 2 x 2 x 17 (or 2² x 17).
9. How are factors of 68 used in finding the Highest Common Factor (HCF)?
To find the HCF of 68 and another number, list the factors of both numbers. The largest factor that appears in both lists is the HCF. For example, to find the HCF of 68 and 34, the factors of 34 are 1, 2, 17, and 34. The HCF is 34.
10. How are the factors of 68 useful in finding the Lowest Common Multiple (LCM)?
The LCM is used to find the smallest number that is a multiple of both numbers. Knowing the factors helps simplify the process. For example, to find the LCM of 68 and another number, understanding the prime factorization of 68 (2² x 17) helps identify common and unique prime factors for calculation.
11. What is the relationship between factors and multiples of 68?
Factors divide a number evenly, while multiples are the result of multiplying a number by whole numbers. For example, the factors of 68 are 1, 2, 4, 17, 34, and 68. Multiples of 68 are 68, 136, 204, etc. They are reciprocally related; if 'a' is a factor of 'b', then 'b' is a multiple of 'a'.
12. Are there any patterns in the factors of 68?
The factors of 68 show that even numbers often have 2 as a factor, and the factors appear in pairs that multiply to give the original number. Analyzing patterns in factors helps improve understanding of number properties and aids in quicker calculations.
