Factors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144. Including 1 and 144 itself, there are 15 distinct factors for 144.
The prime factors of 144 are 2, 3, and its factor pairs are (1, 144), (2, 72), (3, 48), (4, 36), (6, 24), (8, 18), (9, 16), (12, 12). We've put this below in a table for easy sharing.
| Factors of 144: | 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 |
| Prime Factors of 144: | 2, 3 |
| Factor Pairs of 144: | (1, 144), (2, 72), (3, 48), (4, 36), (6, 24), (8, 18), (9, 16), (12, 12) |
How to calculate factors?
To be a factor of 144, a number must divide 144 exactly, leaving no remainder. In other words, when 144 is divided by this number, the quotient is a whole number. These factors, also known as divisors, define the structure of 144 and are key in understanding its mathematical properties.
Below, we outline how to calculate the factorization of 144 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 144:
| Divisor | Is it a factor of 144? | Verification |
|---|---|---|
| 1 | Yes, 1 is a factor of every number. | 1 × 144 = 144 |
| 2 | Yes, 144 is an even number so it's divisible by 2. | 2 × 72 = 144 |
| 3 | Yes, the sum of its digits (9) is divisible by 3. | 3 × 48 = 144 |
| 4 | Yes, the last two digits (44) form a number divisible by 4. | 4 × 36 = 144 |
| 5 | No, last digit is 4, so not divisible by 5. | - |
| 6 | Yes, 144 is divisible by both 2 and 3. | 6 × 24 = 144 |
| 7 | No, 144 divided by 7 leaves a remainder of 4. | - |
| 8 | Yes, the last three digits (144) form a number divisible by 8. | 8 × 18 = 144 |
| 9 | Yes, the sum of its digits (9) is divisible by 9. | 9 × 16 = 144 |
| 10 | No, last digit is 4, so not divisible by 10. | - |
| 11 | No, the difference between sums of alternating digits (1) is not divisible by 11. | - |
| 12 | Yes, 144 is divisible by both 3 and 4. | 12 × 12 = 144 |
| ... | 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 144
You start by dividing 144 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 \( \sqrt144 \)
| Prime Number | Is it a factor of 144? | Verification |
|---|---|---|
| 2 | Yes, 144 is divisible by 2. | 144 ÷ 2 = 72, R0 |
| 2 | Yes, the result 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 144 by performing successive divisions. This method involves dividing 144 by every integer from 1 up to 144 and identifying the numbers that divide exactly without leaving a remainder. In the table below we've ommitted the numbers that don't divide 144, and only kept those that do:
| Divisor | Verification |
|---|---|
| 1 | 144 ÷ 1 = 144 |
| 2 | 144 ÷ 2 = 72 |
| 3 | 144 ÷ 3 = 48 |
| 4 | 144 ÷ 4 = 36 |
| 6 | 144 ÷ 6 = 24 |
| 8 | 144 ÷ 8 = 18 |
| 9 | 144 ÷ 9 = 16 |
| 12 | 144 ÷ 12 = 12 |
| 16 | 144 ÷ 16 = 9 |
| 18 | 144 ÷ 18 = 8 |
| 24 | 144 ÷ 24 = 6 |
| 36 | 144 ÷ 36 = 4 |
| 48 | 144 ÷ 48 = 3 |
| 72 | 144 ÷ 72 = 2 |
| 144 | 144 ÷ 144 = 1 |
Using the division method, we calculated that factors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144.
Factor Tree of 144
The factor tree of 144 shows the step-by-step breakdown of 144 into its prime factors. Each branch of the tree represents a division of 144 into two factors until all resulting factors are prime numbers. This visual representation helps identify the building blocks of 144 and highlights the structure of its prime factorization.
| 144 | |||||
| | | \ | ||||
| 2 | 72 | ||||
| | | \ | ||||
| 2 | 36 | ||||
| | | \ | ||||
| 2 | 18 | ||||
| | | \ | ||||
| 2 | 9 | ||||
| | | \ | ||||
| 3 | 3 |
Factor Pairs of 144 (Visualization)
Factor pairs of 144 are sets of two numbers that, when multiplied together, result in 144. Factor pairs are symmetric and mirror around the square root of 144, such as (1, 144) and (144, 1), and can be both positive and negative pairs as long as their product equals 144.
| Negative factor pairs | Positive factor pairs |
|---|---|
| (-1, -144) | (1, 144) |
| (-2, -72) | (2, 72) |
| (-3, -48) | (3, 48) |
| (-4, -36) | (4, 36) |
| (-6, -24) | (6, 24) |
| (-8, -18) | (8, 18) |
| (-9, -16) | (9, 16) |
| (-12, -12) | (12, 12) |
All factor pairs of 144 are (1, 144), (2, 72), (3, 48), (4, 36), (6, 24), (8, 18), (9, 16), (12, 12), (-1, -144), (-2, -72), (-3, -48), (-4, -36), (-6, -24), (-8, -18), (-9, -16), (-12, -12).
Why Should I Care About Factors of 144?
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 144 bricks and wants to build a wall. If each section of the wall uses 8 bricks, the wall will have 18 sections because 8 × 18 = 144.
- Organizing a Picnic: You have 144 sandwiches and want to split them evenly among your friends. If there are 12 friends, each person will get 12 sandwiches because 12 × 12 = 144.
- Organizing Towels: You have 144 towels to fold and organize. If you stack them in groups of 2, there will be 72 stacks because 2 × 72 = 144.
- Preparing Tables: You have 144 plates to set up for a dinner. If each table needs 4 plates, you will set up 36 tables because 4 × 36 = 144.
- Sharing Gifts: You have 144 gifts to give out at a party. If each guest receives 4 gifts, you’ll have enough for 36 guests because 4 × 36 = 144.