Factors of 75 are 1, 3, 5, 15, 25, 75. Including 1 and 75 itself, there are 6 distinct factors for 75.
The prime factors of 75 are 3, 5, and its factor pairs are (1, 75), (3, 25), (5, 15). We've put this below in a table for easy sharing.
| Factors of 75: | 1, 3, 5, 15, 25, 75 |
| Prime Factors of 75: | 3, 5 |
| Factor Pairs of 75: | (1, 75), (3, 25), (5, 15) |
How to calculate factors?
To be a factor of 75, a number must divide 75 exactly, leaving no remainder. In other words, when 75 is divided by this number, the quotient is a whole number. These factors, also known as divisors, define the structure of 75 and are key in understanding its mathematical properties.
Below, we outline how to calculate the factorization of 75 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 75:
| Divisor | Is it a factor of 75? | Verification |
|---|---|---|
| 1 | Yes, 1 is a factor of every number. | 1 × 75 = 75 |
| 2 | No, 75 is not an even number so it's not divisible by 2. | - |
| 3 | Yes, the sum of its digits (12) is divisible by 3. | 3 × 25 = 75 |
| 4 | No, the last two digits (75) do not form a number divisible by 4. | - |
| 5 | Yes, 75 ends with 0 or 5, so it's divisible by 5. | 5 × 15 = 75 |
| 6 | No, 75 is not divisible by both 2 and 3. | - |
| 7 | No, 75 divided by 7 leaves a remainder of 5. | - |
| 8 | No, the last three digits (75) do not form a number divisible by 8. | - |
| 9 | No, the sum of its digits (12) is not divisible by 9. | - |
| 10 | No, last digit is 5, so not divisible by 10. | - |
| 11 | No, the difference between sums of alternating digits (2) is not divisible by 11. | - |
| 12 | No, 75 is not divisible by both 3 and 4. | - |
| ... | 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 75
You start by dividing 75 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 \( \sqrt75 \)
| Prime Number | Is it a factor of 75? | Verification |
|---|---|---|
| 2 | No, 75 is not divisible by 2. | - |
| 3 | Yes, 75 is divisible by 3. | 75 ÷ 3 = 25, R0 |
| 3 | No, the result 25 is not divisible by 3. | - |
| 5 | Yes, 25 is divisible by 5. | 25 ÷ 5 = 5, R0 |
| 5 | Yes, the result 5 is divisible by 5. | 5 ÷ 5 = 1, R0 |
Method 3: Division Method
The Division Method is a systematic approach to finding all the factors of a 75 by performing successive divisions. This method involves dividing 75 by every integer from 1 up to 75 and identifying the numbers that divide exactly without leaving a remainder. In the table below we've ommitted the numbers that don't divide 75, and only kept those that do:
| Divisor | Verification |
|---|---|
| 1 | 75 ÷ 1 = 75 |
| 3 | 75 ÷ 3 = 25 |
| 5 | 75 ÷ 5 = 15 |
| 15 | 75 ÷ 15 = 5 |
| 25 | 75 ÷ 25 = 3 |
| 75 | 75 ÷ 75 = 1 |
Using the division method, we calculated that factors of 75 are 1, 3, 5, 15, 25, 75.
Factor Tree of 75
The factor tree of 75 shows the step-by-step breakdown of 75 into its prime factors. Each branch of the tree represents a division of 75 into two factors until all resulting factors are prime numbers. This visual representation helps identify the building blocks of 75 and highlights the structure of its prime factorization.
| 75 | ||
| | | \ | |
| 3 | 25 | |
| | | \ | |
| 5 | 5 |
Factor Pairs of 75 (Visualization)
Factor pairs of 75 are sets of two numbers that, when multiplied together, result in 75. Factor pairs are symmetric and mirror around the square root of 75, such as (1, 75) and (75, 1), and can be both positive and negative pairs as long as their product equals 75.
| Negative factor pairs | Positive factor pairs |
|---|---|
| (-1, -75) | (1, 75) |
| (-3, -25) | (3, 25) |
| (-5, -15) | (5, 15) |
All factor pairs of 75 are (1, 75), (3, 25), (5, 15), (-1, -75), (-3, -25), (-5, -15).
Why Should I Care About Factors of 75?
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:
- Hosting a Dinner Party: You have 75 pieces of silverware and want to set the table. If each place setting requires 5 pieces, you can set 15 places because 5 × 15 = 75.
- Arranging Flowers: You have 75 flowers to arrange. If each vase holds 3 flowers, you’ll need 25 vases because 3 × 25 = 75.
- Preparing Tables: You have 75 plates to set up for a dinner. If each table needs 1 plates, you will set up 75 tables because 1 × 75 = 75.
- Creating a Photo Album: You have 75 photos to arrange in an album. If each page holds 1 photos, you’ll fill 75 pages because 1 × 75 = 75.
- Planting a Garden: You have 75 plants to place in a garden. If each row has 5 plants, you’ll create 15 rows because 5 × 15 = 75.