What are the Factors and Prime Factorization of 90?
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: Complete Guide with Lists & Factorization
1. What are the factors of 90?
The factors of 90 are the whole numbers that divide 90 without leaving a remainder. These are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90. Both positive and negative versions of these numbers are considered factors.
2. What are the prime factors of 90?
The prime factorization of 90 expresses it as a product of prime numbers. It is 2 × 3 × 3 × 5, or 2 × 32 × 5. This means 2, 3, and 5 are the prime factors of 90.
3. How many factors does 90 have?
Including both positive and negative factors, 90 has a total of 24 factors. If considering only positive factors, there are 12.
4. What is the factor tree of 90?
A factor tree visually represents the prime factorization. For 90, a possible factor tree would start with 90 = 2 × 45, then 45 = 3 × 15, and finally 15 = 3 × 5. This shows the prime factorization as 2 × 3 × 3 × 5.
5. What are the factor pairs of 90?
Factor pairs are sets of two numbers that multiply to 90. The positive factor pairs are: (1, 90), (2, 45), (3, 30), (5, 18), (6, 15), (9, 10). Negative counterparts of each pair also exist.
6. How do you find the factors of 90?
To find the factors, systematically divide 90 by each integer, starting from 1, until the quotient becomes less than the divisor. If the division results in a whole number (no remainder), the divisor is a factor.
7. What is the sum of the factors of 90?
The sum of all positive factors of 90 is 1 + 2 + 3 + 5 + 6 + 9 + 10 + 15 + 18 + 30 + 45 + 90 = 234
8. What are the common factors of 90 and 60?
Common factors are numbers that divide both 90 and 60 without a remainder. These are 1, 2, 3, 5, 6, 10, 15, and 30.
9. Is 90 a prime or composite number?
90 is a composite number because it has factors other than 1 and itself.
10. How are factors of 90 used in finding the LCM and HCF?
The prime factorization of 90 (2 × 32 × 5) is crucial for calculating the least common multiple (LCM) and highest common factor (HCF) with other numbers. By comparing prime factorizations, you can easily determine the LCM and HCF.
11. Explain the concept of factor pairs in the context of 90.
Factor pairs are two numbers that, when multiplied, equal the given number (90 in this case). For instance, (2, 45) is a factor pair because 2 × 45 = 90. Finding all factor pairs helps in understanding the number's divisibility and structure.
12. Can you explain how to use the division method to find factors?
Divide the number (90) by each integer starting from 1. If the division results in a whole number with no remainder, both the divisor and the quotient are factors of 90. Repeat this process until the quotient becomes smaller than the divisor.
