Factors of 72 – Methods, Pairs, Prime Factorization & Solved Examples

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 = 36

2. 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.