Factors Of 68 And How To Find Them - Vedantu

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 = 34 2. 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:

• Factors of 72
• Prime Factorization
• Factors of a Number
• HCF and LCM
• Divisibility Rules