How to Find All Factors of 90 Step by Step
The concept of factors of 90 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Learning about factors helps you solve questions related to divisibility, prime factorization, LCM, HCF, and more.
What Are Factors of 90?
A factor of 90 is any whole number that divides 90 exactly without leaving a remainder. In mathematics, factors are often used to split numbers into smaller, manageable pieces. You’ll find this concept applied in areas such as finding prime numbers, calculating the LCM and HCF of numbers, and simplifying fractions.
Key Formula for Factors of 90
Here’s the standard way to represent the prime factors of 90:
\( 90 = 2 \times 3^2 \times 5 \)
Complete List and Pair Factors of 90
Factors of 90 include all positive integers that divide 90 exactly. The positive factors are:
| Factor | Pair Factor |
|---|---|
| 1 | 90 |
| 2 | 45 |
| 3 | 30 |
| 5 | 18 |
| 6 | 15 |
| 9 | 10 |
Negative factors are also possible (e.g., -1 × -90 = 90), but in most school-level problems, we focus on positive factors of 90.
How to Find Factors of 90: Step-by-Step
- Start with 1 and continue up to 90. Check each number: does it divide 90 evenly?
- If yes, write down both that number and its pair factor (90 divided by that number).
- List them all in order:
1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 - Stop when you repeat previously found numbers in reverse (since factor pairs are mirrored).
Prime Factorization & Factor Tree of 90
Let’s break 90 into its prime factors. This process is called prime factorization.
- 90 ÷ 2 = 45 (2 is a prime factor)
- 45 ÷ 3 = 15 (3 is a prime factor)
- 15 ÷ 3 = 5 (another 3 is used)
- 5 is a prime number, so we stop.
So, 90 = 2 × 3 × 3 × 5 = 2 × 32 × 5.
Speed Trick or Vedic Shortcut
Here’s a quick tip: if you recall that 90 is divisible by 2, 3, and 5, just use those primes in order. For example, check divisibility by easy rules:
- 90 ends in 0 so it's divisible by 2 and 5.
- Sum the digits: 9 + 0 = 9 → divisible by 3.
Try These Yourself
- List all factors of 90 between 10 and 50.
- Check if 18 is a factor of 90.
- Is 7 a factor of 90? Show your working.
- Find the sum of all positive factors of 90.
Frequent Errors and Misunderstandings
- Forgetting to include 1 and 90 as factors.
- Mixing factor pairs and prime factors.
- Thinking a number is a factor just because it is "close" to 90 (like 8 or 7).
Relation to Other Concepts
The idea of factors of 90 connects closely with concepts like factors and multiples, LCM and HCF, and prime factorization. Mastering this will make calculations with related topics much easier!
Classroom Tip
A simple way to remember factors: draw a factor tree. Build the prime factorization (2, 3, and 5 for 90), then multiply them in all possible combinations to get every factor. Vedantu’s teachers often use this trick onscreen so students can visualize and memorize factor sets easily.
Wrapping It All Up
We explored factors of 90—from definition, formula, steps, and tricks, all the way to relations with LCM and HCF. Practice listing factors of different numbers. For more practice and live explanations, join Vedantu’s interactive lessons and become confident in solving factorization problems quickly.
Explore more about factors and related topics here:
FAQs on Factors of 90 Explained with Prime Factorization
1. What are the factors of 90?
The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90. These are the positive integers that divide 90 exactly without leaving a remainder.
- A factor divides 90 completely.
- For example, 90 ÷ 6 = 15 (no remainder).
- Since 90 is a composite number, it has more than two factors.
2. How do you find the factors of 90?
You can find the factors of 90 by listing numbers that divide 90 exactly or by using prime factorization.
- Step 1: Start dividing 90 by numbers from 1 upward.
- Step 2: Check which numbers leave remainder 0.
- Step 3: List factor pairs such as (1, 90), (2, 45), (3, 30), (5, 18), (6, 15), (9, 10).
- Alternatively, use prime factorization: 90 = 2 × 3² × 5.
3. What is the prime factorization of 90?
The prime factorization of 90 is 2 × 3² × 5. This means 90 is expressed as a product of prime numbers.
- 90 ÷ 2 = 45
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1
4. How many factors does 90 have?
The number 90 has 12 positive factors. Using prime factorization 90 = 2¹ × 3² × 5¹, we apply the formula for total factors.
- Add 1 to each exponent: (1+1)(2+1)(1+1)
- Multiply: 2 × 3 × 2 = 12
5. What are the factor pairs of 90?
The factor pairs of 90 are (1, 90), (2, 45), (3, 30), (5, 18), (6, 15), and (9, 10). Each pair multiplies to give 90.
- 1 × 90 = 90
- 2 × 45 = 90
- 3 × 30 = 90
- 5 × 18 = 90
- 6 × 15 = 90
- 9 × 10 = 90
6. Is 90 a prime or composite number?
The number 90 is a composite number because it has more than two factors. A prime number has exactly two factors: 1 and itself.
- Factors of 90 include 1, 2, 3, 5, and more.
- Since it has 12 factors, it is not prime.
7. What are the common factors of 90 and 60?
The common factors of 90 and 60 are 1, 2, 3, 5, 6, 10, 15, and 30. These numbers divide both 90 and 60 exactly.
- Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
- Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- The overlapping values are the common factors.
8. What is the greatest common factor (GCF) of 90 and 45?
The greatest common factor (GCF) of 90 and 45 is 45. The GCF is the largest number that divides both numbers exactly.
- Factors of 45: 1, 3, 5, 9, 15, 45
- Since 45 divides 90 exactly (90 ÷ 45 = 2), it is the greatest common factor.
9. What is the sum of all factors of 90?
The sum of all positive factors of 90 is 234. Add all its factors to find the total.
- Factors: 1 + 2 + 3 + 5 + 6 + 9 + 10 + 15 + 18 + 30 + 45 + 90
- Total sum = 234
10. What is the smallest and greatest factor of 90?
The smallest factor of 90 is 1 and the greatest factor of 90 is 90. Every positive integer has 1 as its smallest factor and the number itself as its greatest factor.
- 1 divides every number.
- 90 ÷ 90 = 1, so it divides itself exactly.
