Factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90. Including 1 and 90 itself, there are 12 distinct factors for 90.
The prime factors of 90 are 2, 3, 5, and its factor pairs are (1, 90), (2, 45), (3, 30), (5, 18), (6, 15), (9, 10). We've put this below in a table for easy sharing.
| Factors of 90: | 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 |
| Prime Factors of 90: | 2, 3, 5 |
| Factor Pairs of 90: | (1, 90), (2, 45), (3, 30), (5, 18), (6, 15), (9, 10) |
How to calculate factors?
To be a factor of 90, a number must divide 90 exactly, leaving no remainder. In other words, when 90 is divided by this number, the quotient is a whole number. These factors, also known as divisors, define the structure of 90 and are key in understanding its mathematical properties.
Below, we outline how to calculate the factorization of 90 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 90:
| Divisor | Is it a factor of 90? | Verification |
|---|---|---|
| 1 | Yes, 1 is a factor of every number. | 1 × 90 = 90 |
| 2 | Yes, 90 is an even number so it's divisible by 2. | 2 × 45 = 90 |
| 3 | Yes, the sum of its digits (9) is divisible by 3. | 3 × 30 = 90 |
| 4 | No, the last two digits (90) do not form a number divisible by 4. | - |
| 5 | Yes, 90 ends with 0 or 5, so it's divisible by 5. | 5 × 18 = 90 |
| 6 | Yes, 90 is divisible by both 2 and 3. | 6 × 15 = 90 |
| 7 | No, 90 divided by 7 leaves a remainder of 6. | - |
| 8 | No, the last three digits (90) do not form a number divisible by 8. | - |
| 9 | Yes, the sum of its digits (9) is divisible by 9. | 9 × 10 = 90 |
| 10 | Yes, 90 ends with 0, so it's divisible by 10. | 10 × 9 = 90 |
| 11 | No, the difference between sums of alternating digits (9) is not divisible by 11. | - |
| 12 | No, 90 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 90
You start by dividing 90 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 \( \sqrt90 \)
| Prime Number | Is it a factor of 90? | Verification |
|---|---|---|
| 2 | Yes, 90 is divisible by 2. | 90 ÷ 2 = 45, R0 |
| 2 | No, the result 45 is not divisible by 2. | - |
| 3 | Yes, 45 is divisible by 3. | 45 ÷ 3 = 15, R0 |
| 3 | Yes, the result 15 is divisible by 3. | 15 ÷ 3 = 5, R0 |
| 3 | No, the result 5 is not divisible by 3. | - |
| 5 | 5 is a prime number. | 5 is prime. |
Method 3: Division Method
The Division Method is a systematic approach to finding all the factors of a 90 by performing successive divisions. This method involves dividing 90 by every integer from 1 up to 90 and identifying the numbers that divide exactly without leaving a remainder. In the table below we've ommitted the numbers that don't divide 90, and only kept those that do:
| Divisor | Verification |
|---|---|
| 1 | 90 ÷ 1 = 90 |
| 2 | 90 ÷ 2 = 45 |
| 3 | 90 ÷ 3 = 30 |
| 5 | 90 ÷ 5 = 18 |
| 6 | 90 ÷ 6 = 15 |
| 9 | 90 ÷ 9 = 10 |
| 10 | 90 ÷ 10 = 9 |
| 15 | 90 ÷ 15 = 6 |
| 18 | 90 ÷ 18 = 5 |
| 30 | 90 ÷ 30 = 3 |
| 45 | 90 ÷ 45 = 2 |
| 90 | 90 ÷ 90 = 1 |
Using the division method, we calculated that factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.
Factor Tree of 90
The factor tree of 90 shows the step-by-step breakdown of 90 into its prime factors. Each branch of the tree represents a division of 90 into two factors until all resulting factors are prime numbers. This visual representation helps identify the building blocks of 90 and highlights the structure of its prime factorization.
| 90 | |||
| | | \ | ||
| 2 | 45 | ||
| | | \ | ||
| 3 | 15 | ||
| | | \ | ||
| 3 | 5 |
Factor Pairs of 90 (Visualization)
Factor pairs of 90 are sets of two numbers that, when multiplied together, result in 90. Factor pairs are symmetric and mirror around the square root of 90, such as (1, 90) and (90, 1), and can be both positive and negative pairs as long as their product equals 90.
| Negative factor pairs | Positive factor pairs |
|---|---|
| (-1, -90) | (1, 90) |
| (-2, -45) | (2, 45) |
| (-3, -30) | (3, 30) |
| (-5, -18) | (5, 18) |
| (-6, -15) | (6, 15) |
| (-9, -10) | (9, 10) |
All factor pairs of 90 are (1, 90), (2, 45), (3, 30), (5, 18), (6, 15), (9, 10), (-1, -90), (-2, -45), (-3, -30), (-5, -18), (-6, -15), (-9, -10).
Why Should I Care About Factors of 90?
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:
- Splitting Pizza Slices: If you have 90 slices of pizza and 1 friends, each friend can have 90 slices, since 1 × 90 = 90.
- Hosting a Party: You have 90 party favors and want to give each guest the same amount. If there are 9 guests, each guest gets 10 party favors because 9 × 10 = 90.
- Arranging Chairs for a Concert: You have 90 chairs for a concert. If you arrange them into 1 rows, each row will have 90 chairs because 1 × 90 = 90.
- Making Cookies: You have 90 cookie dough balls to bake. If each tray holds 5 dough balls, you’ll need 18 trays because 5 × 18 = 90.
- Organizing School Desks: You have 90 desks to arrange in a classroom. If each row has 5 desks, you’ll create 18 rows because 5 × 18 = 90.
