Factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. Including 1 and 72 itself, there are 12 distinct factors for 72.
The prime factors of 72 are 2, 3, and its factor pairs are (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), (8, 9). We've put this below in a table for easy sharing.
| Factors of 72: | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 |
| Prime Factors of 72: | 2, 3 |
| Factor Pairs of 72: | (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), (8, 9) |
How to calculate factors?
To be a factor of 72, a number must divide 72 exactly, leaving no remainder. In other words, when 72 is divided by this number, the quotient is a whole number. These factors, also known as divisors, define the structure of 72 and are key in understanding its mathematical properties.
Below, we outline how to calculate the factorization of 72 using four methods: basic factorization, prime factorization, the division method, or using GCD and LCM. We also include a detailed analysis of factor pairs and a factor tree to illustrate the breakdown.
Method 1: Basic Factorization
Basic Factorization is a method to find the factors of a number by systematically testing each whole number from 2 up to the number itself to see which ones divides with zero remainder (evenly). The process is somewhat time consuming if a number is high, that's why you should master divisibility rules, to make the process faster.
Here's the breakdown for 72:
| Divisor | Is it a factor of 72? | Verification |
|---|---|---|
| 1 | Yes, 1 is a factor of every number. | 1 × 72 = 72 |
| 2 | Yes, 72 is an even number so it's divisible by 2. | 2 × 36 = 72 |
| 3 | Yes, the sum of its digits (9) is divisible by 3. | 3 × 24 = 72 |
| 4 | Yes, the last two digits (72) form a number divisible by 4. | 4 × 18 = 72 |
| 5 | No, last digit is 2, so not divisible by 5. | - |
| 6 | Yes, 72 is divisible by both 2 and 3. | 6 × 12 = 72 |
| 7 | No, 72 divided by 7 leaves a remainder of 2. | - |
| 8 | Yes, the last three digits (72) form a number divisible by 8. | 8 × 9 = 72 |
| 9 | Yes, the sum of its digits (9) is divisible by 9. | 9 × 8 = 72 |
| 10 | No, last digit is 2, so not divisible by 10. | - |
| 11 | No, the difference between sums of alternating digits (5) is not divisible by 11. | - |
| 12 | Yes, 72 is divisible by both 3 and 4. | 12 × 6 = 72 |
| ... | continue with all the other numbers. |
Method 2: Prime Factorization
Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. This means a prime number cannot be formed by multiplying two smaller natural numbers. For example:
- 2 is a prime number because its only divisors are 1 and 2.
- 3 is prime for the same reason—it can only be divided evenly by 1 and 3.
- 4 is not prime because it can be divided by 1, 2, and 4.
- 5, 7, 11, and 13 are also prime numbers.
Prime numbers are fundamental in mathematics because they are the "building blocks" of whole numbers. Any natural number greater than 1 can be expressed as a product of prime numbers, which is known as its prime factorization.
How to do prime factorization of 72
You start by dividing 72 to each prime number, multiple times, until the remainder is 0. Then you move on to the next prime number. To save time, you should test with up to \( \sqrt72 \)
| Prime Number | Is it a factor of 72? | Verification |
|---|---|---|
| 2 | Yes, 72 is divisible by 2. | 72 ÷ 2 = 36, R0 |
| 2 | Yes, the result 36 is divisible by 2. | 36 ÷ 2 = 18, R0 |
| 2 | Yes, the result 18 is divisible by 2. | 18 ÷ 2 = 9, R0 |
| 2 | No, the result 9 is not divisible by 2. | - |
| 3 | Yes, 9 is divisible by 3. | 9 ÷ 3 = 3, R0 |
| 3 | Yes, the result 3 is divisible by 3. | 3 ÷ 3 = 1, R0 |
Method 3: Division Method
The Division Method is a systematic approach to finding all the factors of a 72 by performing successive divisions. This method involves dividing 72 by every integer from 1 up to 72 and identifying the numbers that divide exactly without leaving a remainder. In the table below we've ommitted the numbers that don't divide 72, and only kept those that do:
| Divisor | Verification |
|---|---|
| 1 | 72 ÷ 1 = 72 |
| 2 | 72 ÷ 2 = 36 |
| 3 | 72 ÷ 3 = 24 |
| 4 | 72 ÷ 4 = 18 |
| 6 | 72 ÷ 6 = 12 |
| 8 | 72 ÷ 8 = 9 |
| 9 | 72 ÷ 9 = 8 |
| 12 | 72 ÷ 12 = 6 |
| 18 | 72 ÷ 18 = 4 |
| 24 | 72 ÷ 24 = 3 |
| 36 | 72 ÷ 36 = 2 |
| 72 | 72 ÷ 72 = 1 |
Using the division method, we calculated that factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
Factor Tree of 72
The factor tree of 72 shows the step-by-step breakdown of 72 into its prime factors. Each branch of the tree represents a division of 72 into two factors until all resulting factors are prime numbers. This visual representation helps identify the building blocks of 72 and highlights the structure of its prime factorization.
| 72 | ||||
| | | \ | |||
| 2 | 36 | |||
| | | \ | |||
| 2 | 18 | |||
| | | \ | |||
| 2 | 9 | |||
| | | \ | |||
| 3 | 3 |
Factor Pairs of 72 (Visualization)
Factor pairs of 72 are sets of two numbers that, when multiplied together, result in 72. Factor pairs are symmetric and mirror around the square root of 72, such as (1, 72) and (72, 1), and can be both positive and negative pairs as long as their product equals 72.
| Negative factor pairs | Positive factor pairs |
|---|---|
| (-1, -72) | (1, 72) |
| (-2, -36) | (2, 36) |
| (-3, -24) | (3, 24) |
| (-4, -18) | (4, 18) |
| (-6, -12) | (6, 12) |
| (-8, -9) | (8, 9) |
All factor pairs of 72 are (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), (8, 9), (-1, -72), (-2, -36), (-3, -24), (-4, -18), (-6, -12), (-8, -9).
Why Should I Care About Factors of 72?
Turns out, factors aren’t just about boring math equations—they’re like secret superpowers hiding inside numbers! Knowing them can help you split things up, share with friends, or even spot hidden patterns. Want to know how? Check out these real-life examples that show just how cool factors really are:
- Building a Wall: A builder has 72 bricks and wants to build a wall. If each section of the wall uses 1 bricks, the wall will have 72 sections because 1 × 72 = 72.
- Creating Gift Bags: You want to make gift bags and have 72 gifts. If you put 3 gifts in each bag, you’ll make 24 bags because 3 × 24 = 72.
- Setting Up Tents: You have 72 tents and want to divide campers into groups. If each group uses 4 tents, there will be 18 groups because 4 × 18 = 72.
- Making Bracelets: You have 72 beads and want to make bracelets. If each bracelet requires 4 beads, you can make 18 bracelets because 4 × 18 = 72.
- Filling a Jar with Candy: You have 72 candies to put in a jar. If each jar holds 4 candies, you’ll fill 18 jars because 4 × 18 = 72.
