Unlocking the Secrets of 90: A Comprehensive Guide to its Factors
Understanding factors is a fundamental concept in mathematics, essential for grasping more advanced topics like prime factorization, fractions, and algebra. This article will delve into the factors of 90, a seemingly simple number that offers a rich opportunity to explore this important mathematical idea. We'll break down the process step-by-step, using clear explanations and practical examples to make the learning experience engaging and accessible.
What are Factors?
A factor is a number that divides another number exactly, without leaving a remainder. In simpler terms, if you can divide a number by another number and get a whole number result, then the number you divided by is a factor. For example, the factors of 6 are 1, 2, 3, and 6, because 6 can be divided evenly by each of these numbers. We can represent this as 6 ÷ 1 = 6, 6 ÷ 2 = 3, 6 ÷ 3 = 2, and 6 ÷ 6 = 1.
Finding the Factors of 90: A Systematic Approach
To find all the factors of 90, we can use a methodical approach. We start by checking each number, beginning from 1, to see if it divides 90 without leaving a remainder.
1. Start with 1: Every number has 1 as a factor. 90 ÷ 1 = 90. So, 1 and 90 are factors.
2. Check for 2: 90 is an even number, so it's divisible by 2. 90 ÷ 2 = 45. Thus, 2 and 45 are factors.
3. Check for 3: The sum of the digits of 90 (9 + 0 = 9) is divisible by 3, indicating that 90 is divisible by 3. 90 ÷ 3 = 30. So, 3 and 30 are factors.
4. Check for 4: 90 is not divisible by 4 because it doesn't end in a multiple of 4 (00, 04, 08, etc.).
5. Check for 5: 90 ends in 0, so it's divisible by 5. 90 ÷ 5 = 18. Hence, 5 and 18 are factors.
6. Check for 6: Since 90 is divisible by both 2 and 3, it's also divisible by 6. 90 ÷ 6 = 15. Therefore, 6 and 15 are factors.
7. Check for 7: 90 ÷ 7 = 12 with a remainder, so 7 is not a factor.
8. Check for 8: 90 is not divisible by 8.
9. Check for 9: 90 is divisible by 9 because 90 ÷ 9 = 10. Thus, 9 and 10 are factors.
10. Check for 10: We already found that 10 is a factor.
We've now systematically checked numbers up to 10. Notice that as we progress, we start finding factor pairs. Once we reach a factor (e.g., 10) whose pair (9) we’ve already found, we can stop. We’ve found all the factors.
The Complete List of Factors of 90
Therefore, the factors of 90 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.
Practical Applications of Factors
Understanding factors is crucial in many real-life situations:
Dividing objects equally: Imagine you have 90 candies to distribute equally among friends. You can find out how many friends you can give candies to by considering the factors of 90.
Simplifying fractions: Factors are essential for simplifying fractions to their lowest terms. For example, the fraction 45/90 can be simplified using the common factor 45, reducing it to 1/2.
Solving algebraic equations: Finding factors is crucial in solving many types of algebraic equations.
Key Takeaways
Factors are whole numbers that divide a number without leaving a remainder.
A systematic approach helps identify all factors efficiently.
Understanding factors is fundamental to various mathematical concepts and real-world applications.
Frequently Asked Questions (FAQs)
1. What is the prime factorization of 90? The prime factorization of 90 is 2 x 3² x 5. This means 90 is composed of only prime numbers multiplied together.
2. How many factors does 90 have? 90 has 12 factors.
3. Are all factors of 90 even numbers? No, several factors of 90 are odd numbers (1, 3, 5, 9, 15, 45).
4. Is there a quicker method to find factors than the one described? While the systematic approach is reliable, more advanced techniques like factor trees and prime factorization can be used for larger numbers to find factors more quickly.
5. Why is understanding factors important in higher-level mathematics? Factors are essential for understanding concepts like greatest common divisor (GCD), least common multiple (LCM), and polynomial factorization, all crucial in algebra and number theory.
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