How to Find Factors and Prime Factorization of 86
The concept of factors of 86 is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Knowing how to find factors, factor pairs, and the prime factors of a number like 86 improves number sense and supports many maths topics, including divisibility, multiples, HCF, and LCM.
Understanding Factors of 86
A factor of 86 is a whole number that divides 86 exactly without leaving any remainder. Factors are crucial in the study of divisors, multiples, and prime factorization, and are often used in both arithmetic and algebra. Learning about factors of numbers like 86 helps with topics such as highest common factor (HCF), lowest common multiple (LCM), and understanding the building blocks of composite and prime numbers.
How to Find Factors of 86
To find the factors of 86, follow these simple steps:
1. Start with the smallest whole number, 1.
2. Check if 1 divides 86 without a remainder. \( 86 \div 1 = 86 \), so 1 is a factor.
3. Move to the next whole number. \( 86 \div 2 = 43 \) with no remainder, so 2 is a factor.
4. Try 3. \( 86 \div 3 = 28.666... \) (not a whole number), so 3 is not a factor.
5. Continue checking up to \( \sqrt{86} \) (around 9.27).
6. Next, try 43. \( 86 \div 43 = 2 \), so 43 is a factor.
7. 86 divided by itself gives 1 (i.e., \( 86 \div 86 = 1 \)), so 86 is a factor.
There are no other whole numbers that divide 86 exactly. So, the complete list is below.
List of All Factors of 86
The whole number factors of 86, in order from smallest to largest, are:
1, 2, 43, 86
These numbers divide 86 completely, leaving no remainder.
Factors of 86 in Pairs (Factor Pairs)
Factor pairs are two numbers that multiply to give 86. Let’s see them in a table:
Factor Pairs of 86
| Factor 1 | Factor 2 | Product |
|---|---|---|
| 1 | 86 | 1 × 86 = 86 |
| 2 | 43 | 2 × 43 = 86 |
So, the factor pairs of 86 are (1, 86) and (2, 43).
Prime Factorization of 86
Prime factorization means expressing 86 as a product of its prime factors. Let’s do it step by step:
1. 86 is an even number. Start dividing by 2 (the smallest prime):
2. Now, check if 43 is prime. 43 is a prime number.
3. So, the prime factorization of 86 is:
A simple factor tree would also show: 86 splits to 2 and 43. Both are now prime.
Division Method to Find Factors of 86
Using the division method, we check if 86 is divisible by whole numbers:
- \( 86 \div 1 = 86 \) (factor)
- \( 86 \div 2 = 43 \) (factor)
- \( 86 \div 3 = 28.67... \) (not a factor)
- \( 86 \div 4 = 21.5 \) (not a factor)
- ...
- \( 86 \div 43 = 2 \) (factor)
- \( 86 \div 86 = 1 \) (factor)
All other numbers between 1 and 86 either leave a remainder or don’t result in whole numbers.
Worked Example – Solving a Problem
Let’s find the sum and average of all factors of 86:
1. List the factors: 1, 2, 43, 86
2. Add them: \( 1 + 2 + 43 + 86 = 132 \)
3. Find how many factors: 4
4. Calculate the average: \( 132 \div 4 = 33 \)
Related Numbers – Factors of 85 and 87
It helps to compare with nearby numbers:
- Factors of 85: 1, 5, 17, 85
- Factors of 87: 1, 3, 29, 87
See how each number has a different set of pairs and divisors. For more examples, see Factors of 85 and Factors of 87.
Practice Problems
- List all factors of 86.
- What are the factor pairs of 86?
- What is the prime factorization of 86?
- Is 43 a factor of 86? Explain.
- Find the HCF of 42 and 86 using their factors.
Common Mistakes to Avoid
- Including numbers like 3, 4, or 5, which do not divide 86 exactly.
- Confusing factors with multiples (multiples of 86 are 86, 172, 258,..., but not factors!)
- Forgetting that 1 and 86 are always factors of 86.
- Thinking 86 is prime; it is composite since it has more than two factors.
Real-World Applications
The concept of factors of 86 appears in dividing objects into equal parts, setting up groups/lots, understanding divisibility in maths, and simplifying calculations for LCM or HCF. Vedantu helps students see how factoring skills are useful for exams as well as daily problem-solving.
We explored the idea of factors of 86, listed their pairs, solved examples, and saw related real-life uses. Practice more such problems on Vedantu to strengthen your understanding of factors, multiples, and primes for board exams and beyond!
Also discover more with these helpful pages: Prime Numbers, Table of 43 (prime factor), Prime Factorization, and learn the foundation with the Fundamental Theorem of Arithmetic.
FAQs on Factors of 86 Explained with Examples
1. What are the factors of 86?
The factors of 86 are the numbers that divide 86 exactly without leaving a remainder. These are 1, 2, 43, and 86. Each of these numbers multiplied by its pair gives the original number 86.
2. Is 86 a prime or composite number?
86 is a composite number because it has more than two factors. A prime number has exactly two factors (1 and itself), whereas 86 has four factors: 1, 2, 43, and 86.
3. How do you make a factor tree for 86?
To create a factor tree for 86, start by dividing 86 by its smallest prime factor, which is 2. Then, divide the quotient by its prime factors until all leaves in the tree are prime numbers. For 86, the factor tree shows 86 breaks down into 2 and 43, both prime numbers.
4. What is the prime factorization of 86?
The prime factorization of 86 is the expression of 86 as a product of its prime factors. Since 86 = 2 × 43, and both 2 and 43 are prime numbers, its prime factorization is 2 × 43.
5. What is the HCF of 86 and 42?
The HCF (Highest Common Factor) of 86 and 42 is the greatest number that divides both without remainder. The factors of 86 are 1, 2, 43, and 86; the factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. The common factors are 1 and 2, so the HCF is 2.
6. How many factor pairs does 86 have?
86 has two factor pairs whose product is 86: (1, 86) and (2, 43). Both pairs multiply to give 86, illustrating how factors come in pairs.
7. Why is 43 considered a prime factor of 86?
43 is considered a prime factor of 86 because it is a prime number and it divides 86 exactly without a remainder. Since no other number besides 1 and 43 divides 43, it qualifies as a prime factor in the multiplication that forms 86.
8. Why do students confuse factors and multiples of 86?
Students often confuse factors and multiples because both relate to division and multiplication. Factors of 86 are numbers that divide 86 exactly, while multiples of 86 are numbers obtained by multiplying 86 by whole numbers. Understanding this difference helps in solving related problems accurately.
9. Can 86 have decimal or negative factors?
While factors are typically considered as whole numbers, negative factors of 86 do exist because if two negatives multiply to give a positive number, they are factor pairs of 86, such as (-1, -86) and (-2, -43). However, decimal factors are not counted as factors since factors are whole numbers dividing the number exactly without a remainder.
10. Is there a shortcut to list factors in exams?
Yes, a common shortcut to list factors quickly includes:
- Start dividing the number by small prime numbers like 2, 3, 5, etc.
- Note the divisor and quotient as factor pairs whenever the division leaves no remainder.
- Stop once the divisor exceeds the square root of the number.
11. How does factorization help in finding LCM or HCF?
Factorization helps in finding the LCM (Least Common Multiple) and HCF (Highest Common Factor) by breaking numbers into their prime factors. For HCF, common prime factors with the smallest powers are multiplied. For LCM, all prime factors with the highest powers are multiplied. This method simplifies calculation and is especially useful in exam settings.
12. What are the factors of 86 by the division method?
Using the division method, factors of 86 are found by dividing 86 by numbers from 1 up to 86 and checking for no remainder. The divisions that result in whole numbers are:
86 ÷ 1 = 86
86 ÷ 2 = 43
86 ÷ 43 = 2
86 ÷ 86 = 1
Thus, the factors of 86 are 1, 2, 43, and 86.
