What are the Factors and Factor Pairs of 72?
The concept of factors of 72 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Factors help us break down numbers, find common divisors, and solve many arithmetic and algebraic problems quickly.
What Are Factors of 72?
The factors of 72 are all whole numbers that can divide 72 exactly, leaving no remainder. This means if you multiply any two of these numbers together (in a certain pair), you will get exactly 72. You’ll find this concept applied in areas such as division, HCF and LCM problems, and quickly solving questions in competitive exams or class tests.
List of all factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
Pair Factors of 72
Here are the factor pairs of 72, where each pair multiplies to give 72:
| Factor 1 | Factor 2 |
|---|---|
| 1 | 72 |
| 2 | 36 |
| 3 | 24 |
| 4 | 18 |
| 6 | 12 |
| 8 | 9 |
Key Formula for Factors of 72
There’s no formula for all factors, but for prime factorization of 72, use: \( 72 = 2^3 \times 3^2 \).
Prime Factorization of 72
The prime factorization is the process of breaking 72 into its basic prime number multipliers. Here’s how you do it:
1. Divide 72 by 2: 72 ÷ 2 = 362. Divide 36 by 2: 36 ÷ 2 = 18
3. Divide 18 by 2: 18 ÷ 2 = 9
4. Divide 9 by 3: 9 ÷ 3 = 3
5. Divide 3 by 3: 3 ÷ 3 = 1
So, prime factorization of 72 is 2 × 2 × 2 × 3 × 3 or written as 23 × 32.
How to Find Factors of 72 – Step-by-Step
Let’s use the division method to check which numbers divide 72:
1. Start with 1: 72 ÷ 1 = 72 (factor)2. Try 2: 72 ÷ 2 = 36 (factor)
3. Try 3: 72 ÷ 3 = 24 (factor)
4. Try 4: 72 ÷ 4 = 18 (factor)
5. Try 5: 72 ÷ 5 = 14.4 (not a factor)
6. Continue this up to 72, listing all even divisions. The ones with no remainder are your factors.
You can stop once your division result starts repeating numbers (for 8, you get 9; for 9, you get 8).
Speed Trick or Vedic Shortcut
Instead of testing all numbers, check the prime factors first. For composite numbers like 72, break it down into primes (2 and 3) and generate all possible products from their powers. This helps quickly list factors in exams.
Example Trick: Start with the lowest prime, keep dividing, and build a factor tree. This also speeds up HCF/LCM problems.
Try These Yourself
- Write all the factors of 72 in order.
- Find the factors of 36 and compare them with factors of 72.
- Check if 24 is a factor of 72.
- List all even factors of 72.
- Find the sum of all the factors of 72.
Frequent Errors and Misunderstandings
- Confusing factors with multiples. Remember, factors divide the number exactly; multiples are found by multiplying the number.
- Forgetting that '1' and the number itself (here, 72) are always factors.
- Assuming only even numbers can be factors of an even number like 72 (but 3 and 9 are factors, and they are odd).
Relation to Other Concepts
Knowing the factors of 72 helps when learning about LCM and HCF and when exploring prime numbers and divisibility rules. Factors also appear in algebra, ratios, fractions, and more. The skill is useful from Class 4 up to competitive exams.
Classroom Tip
A quick way to remember factors of 72 is to pair up numbers—start from 1 and find the number you multiply with to get 72 (like 1×72, 2×36, etc.). Forming these pairs helps visualize factors and doubles as a memory card for revision. Vedantu’s teachers also use tables and factor trees during live classes to make learning interactive.
We explored factors of 72—from the definition, pair lists, prime factorization, tricks, errors, and how this fits with topics like HCF/LCM and prime numbers. Keep practicing with Vedantu’s free resources and get comfortable with factorization for a strong Math foundation.
Check out these related topics: Prime Factorization, Factors of 48, Factors of 36, LCM and HCF.
FAQs on Factors of 72 – Methods, Pairs, Prime Factorization & Solved Examples
1. What are the factors of 72?
The factors of 72 are all whole numbers that divide 72 exactly, leaving no remainder. These factors include:
- 1
- 2
- 3
- 4
- 6
- 8
- 9
- 12
- 18
- 24
- 36
- 72
2. What is the factor tree of 72?
A factor tree visually breaks down 72 into its prime factors step by step. The factor tree for 72 is:
- Start with 72.
- 72 can be divided by 2: $72 = 2 \times 36$
- Break down 36: $36 = 2 \times 18$
- Break down 18: $18 = 2 \times 9$
- Break down 9: $9 = 3 \times 3$
3. What are the multiples of 72?
The multiples of 72 are numbers that can be expressed as $72 \times n$, where n is a natural number. Some of the first few multiples are:
- 72 $\times$ 1 = 72
- 72 $\times$ 2 = 144
- 72 $\times$ 3 = 216
- 72 $\times$ 4 = 288
- 72 $\times$ 5 = 360
4. Why is 3 a factor of 72?
The number 3 is a factor of 72 because 72 can be divided by 3 without any remainder. Specifically, $72 \div 3 = 24$, which is a whole number. In mathematical terms, since $3 \times 24 = 72$, 3 is considered a factor. Understanding factors is foundational for advanced topics, and Vedantu’s math tutors use practical examples to reinforce such concepts in every session.
5. What is the prime factorization of 72?
The prime factorization of 72 breaks it down into its smallest prime number components. Using repeated division:
- 72 $\div$ 2 = 36
- 36 $\div$ 2 = 18
- 18 $\div$ 2 = 9
- 9 $\div$ 3 = 3
- 3 $\div$ 3 = 1
6. How do you find all pairs of factors of 72?
To find all pairs of factors for 72, list both numbers in each multiplication combination that equals 72. The factor pairs are:
- 1 and 72
- 2 and 36
- 3 and 24
- 4 and 18
- 6 and 12
- 8 and 9
7. Is 72 a perfect square or a perfect cube?
A perfect square is a number that can be written as $n^2$. For 72, $\sqrt{72} \approx 8.485$ which is not an integer, so 72 is not a perfect square.
A perfect cube is a number that can be written as $n^3$. $\sqrt[3]{72} \approx 4.16$, also not an integer. Therefore, 72 is not a perfect cube. Vedantu’s comprehensive learning modules help clarify such concepts with real-life applications and step-by-step problem solving.
8. What is the sum and difference of the factors of 72?
The sum of all factors of 72 is calculated by adding each unique factor: $1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 36 + 72 = 195$.
The difference between the largest and smallest factor is $72 - 1 = 71$. Learning how to calculate sums and differences of factors is made easy with Vedantu’s systematic teaching approach and detailed worksheets.
9. How can understanding factors of 72 help in solving LCM and HCF problems?
Understanding the factors of 72 is essential for finding the Lowest Common Multiple (LCM) and Highest Common Factor (HCF) in mathematics. Knowing 72’s prime factors ($2^3 \times 3^2$) lets students:
- Easily calculate LCM and HCF with other numbers
- Simplify fractions
- Solve divisibility problems
