Factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Including 1 and 48 itself, there are 10 distinct factors for 48.
The prime factors of 48 are 2, 3, and its factor pairs are (1, 48), (2, 24), (3, 16), (4, 12), (6, 8). We've put this below in a table for easy sharing.
| Factors of 48: | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 |
| Prime Factors of 48: | 2, 3 |
| Factor Pairs of 48: | (1, 48), (2, 24), (3, 16), (4, 12), (6, 8) |
How to calculate factors?
To be a factor of 48, a number must divide 48 exactly, leaving no remainder. In other words, when 48 is divided by this number, the quotient is a whole number. These factors, also known as divisors, define the structure of 48 and are key in understanding its mathematical properties.
Below, we outline how to calculate the factorization of 48 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 48:
| Divisor | Is it a factor of 48? | Verification |
|---|---|---|
| 1 | Yes, 1 is a factor of every number. | 1 × 48 = 48 |
| 2 | Yes, 48 is an even number so it's divisible by 2. | 2 × 24 = 48 |
| 3 | Yes, the sum of its digits (12) is divisible by 3. | 3 × 16 = 48 |
| 4 | Yes, the last two digits (48) form a number divisible by 4. | 4 × 12 = 48 |
| 5 | No, last digit is 8, so not divisible by 5. | - |
| 6 | Yes, 48 is divisible by both 2 and 3. | 6 × 8 = 48 |
| 7 | No, 48 divided by 7 leaves a remainder of 6. | - |
| 8 | Yes, the last three digits (48) form a number divisible by 8. | 8 × 6 = 48 |
| 9 | No, the sum of its digits (12) is not divisible by 9. | - |
| 10 | No, last digit is 8, so not divisible by 10. | - |
| 11 | No, the difference between sums of alternating digits (4) is not divisible by 11. | - |
| 12 | Yes, 48 is divisible by both 3 and 4. | 12 × 4 = 48 |
| ... | 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 48
You start by dividing 48 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 \( \sqrt48 \)
| Prime Number | Is it a factor of 48? | Verification |
|---|---|---|
| 2 | Yes, 48 is divisible by 2. | 48 ÷ 2 = 24, R0 |
| 2 | Yes, the result 24 is divisible by 2. | 24 ÷ 2 = 12, R0 |
| 2 | Yes, the result 12 is divisible by 2. | 12 ÷ 2 = 6, R0 |
| 2 | Yes, the result 6 is divisible by 2. | 6 ÷ 2 = 3, R0 |
| 2 | No, the result 3 is not divisible by 2. | - |
| 3 | 3 is a prime number. | 3 is prime. |
Method 3: Division Method
The Division Method is a systematic approach to finding all the factors of a 48 by performing successive divisions. This method involves dividing 48 by every integer from 1 up to 48 and identifying the numbers that divide exactly without leaving a remainder. In the table below we've ommitted the numbers that don't divide 48, and only kept those that do:
| Divisor | Verification |
|---|---|
| 1 | 48 ÷ 1 = 48 |
| 2 | 48 ÷ 2 = 24 |
| 3 | 48 ÷ 3 = 16 |
| 4 | 48 ÷ 4 = 12 |
| 6 | 48 ÷ 6 = 8 |
| 8 | 48 ÷ 8 = 6 |
| 12 | 48 ÷ 12 = 4 |
| 16 | 48 ÷ 16 = 3 |
| 24 | 48 ÷ 24 = 2 |
| 48 | 48 ÷ 48 = 1 |
Using the division method, we calculated that factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
Factor Tree of 48
The factor tree of 48 shows the step-by-step breakdown of 48 into its prime factors. Each branch of the tree represents a division of 48 into two factors until all resulting factors are prime numbers. This visual representation helps identify the building blocks of 48 and highlights the structure of its prime factorization.
| 48 | ||||
| | | \ | |||
| 2 | 24 | |||
| | | \ | |||
| 2 | 12 | |||
| | | \ | |||
| 2 | 6 | |||
| | | \ | |||
| 2 | 3 |
Factor Pairs of 48 (Visualization)
Factor pairs of 48 are sets of two numbers that, when multiplied together, result in 48. Factor pairs are symmetric and mirror around the square root of 48, such as (1, 48) and (48, 1), and can be both positive and negative pairs as long as their product equals 48.
| Negative factor pairs | Positive factor pairs |
|---|---|
| (-1, -48) | (1, 48) |
| (-2, -24) | (2, 24) |
| (-3, -16) | (3, 16) |
| (-4, -12) | (4, 12) |
| (-6, -8) | (6, 8) |
All factor pairs of 48 are (1, 48), (2, 24), (3, 16), (4, 12), (6, 8), (-1, -48), (-2, -24), (-3, -16), (-4, -12), (-6, -8).
Why Should I Care About Factors of 48?
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:
- Watering Plants: You have 48 plants that need watering. If each watering can waters 2 plants, you’ll need 24 cans to water them all because 2 × 24 = 48.
- Handing Out Treats: You have 48 cookies and want to distribute them equally. If each person gets 2 cookies, you can serve 24 people because 2 × 24 = 48.
- Organizing A Car Wash: You have 48 cars to wash. If each team can wash 4 cars, you’ll need 12 teams because 4 × 12 = 48.
- Making Cookies: You have 48 cookie dough balls to bake. If each tray holds 2 dough balls, you’ll need 24 trays because 2 × 24 = 48.
- Creating a Puzzle: You have 48 pieces to make a puzzle. If each section has 6 pieces, you’ll have 8 sections because 6 × 8 = 48.
