The highest common factor (HCF) of 36 and 90 is 18.
Understanding the Highest Common Factor (HCF)
The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest positive integer that divides two or more numbers without leaving a remainder. It's a fundamental concept in number theory, crucial for simplifying fractions and solving various mathematical problems.
How to Find the HCF of 36 and 90
To determine the HCF of 36 and 90, we follow a systematic approach:
- List all factors: Identify all the numbers that divide each of the given numbers (36 and 90) exactly.
- Identify common factors: Find the factors that appear in the lists for both numbers.
- Select the highest: From the common factors, choose the largest one.
Let's break down the process for 36 and 90:
Factors of 36 and 90
Here are the factors for each number:
| Number | Factors |
|---|---|
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 |
| 90 | 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 |
Identifying the Highest Common Factor
By examining the factors of both 36 and 90, we can identify their common factors:
- Common Factors: 1, 2, 3, 6, 9, 18
Among these common factors, the highest value is 18. Therefore, 18 is the Highest Common Factor of 36 and 90.
Alternative Method: Prime Factorization
Another powerful method to find the HCF is through prime factorization. This involves breaking down each number into its prime factors:
- Prime Factors of 36: $2 \times 2 \times 3 \times 3$ or $2^2 \times 3^2$
- Prime Factors of 90: $2 \times 3 \times 3 \times 5$ or $2^1 \times 3^2 \times 5^1$
To find the HCF using prime factorization, you take the lowest power of each common prime factor:
- Common prime factors are 2 and 3.
- Lowest power of 2: $2^1$ (from 90's factors)
- Lowest power of 3: $3^2$ (common to both)
Multiplying these gives: $2^1 \times 3^2 = 2 \times 9 = 18$.
For more information on calculating HCF, you can explore resources like Khan Academy's HCF explanations or Wikipedia's page on the Greatest Common Divisor.
Practical Applications of HCF
The HCF is not just a theoretical concept; it has several practical applications:
- Simplifying Fractions: The HCF is used to reduce fractions to their simplest form. For example, to simplify 36/90, you would divide both the numerator and the denominator by their HCF (18), resulting in 2/5.
- Dividing Items into Groups: In real-world scenarios, the HCF helps in dividing a set of items into the largest possible equal-sized groups without any leftovers.
- Solving Word Problems: Many word problems involve finding the largest possible size or group, directly applying the HCF concept.
