Highest Common Factor Of 12 And 24: Quick Guide
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To find the Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two numbers, in this case, 12 and 24, involves understanding the concept of factors and how to identify the largest one shared between them. This guide will provide a comprehensive overview of how to determine the HCF of these two numbers, with clear explanations and practical examples.
Understanding Factors
Factors are numbers that divide another number completely, meaning without leaving any remainder. For example:
- The factors of 12 are:
- 1, 2, 3, 4, 6, 12
- The factors of 24 are:
- 1, 2, 3, 4, 6, 8, 12, 24
To find the HCF, we will identify the common factors between these two sets and then determine the highest one.
Step-by-Step Process to Find HCF
1. List the Factors
As already noted, the factors of 12 and 24 are:
| Number | Factors |
|---|---|
| 12 | 1, 2, 3, 4, 6, 12 |
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 |
2. Identify Common Factors
Now, let’s find the common factors by comparing the two sets:
- Common factors of 12 and 24:
- 1, 2, 3, 4, 6, 12
3. Determine the Highest Common Factor
Among the common factors listed, the largest number is 12. Therefore, the HCF of 12 and 24 is 12.
Alternative Methods to Find HCF
There are different methods to calculate the HCF:
Method 1: Prime Factorization
Using prime factorization is another effective way to determine the HCF:
- Prime Factorization of 12:
- 12 = 2² × 3¹
- Prime Factorization of 24:
- 24 = 2³ × 3¹
To find the HCF using prime factorization:
- Take the lowest power of each common prime factor.
- Here, the common primes are 2 and 3.
- For 2, the lowest power is 2² (from 12).
- For 3, the lowest power is 3¹.
Now, we multiply these together:
[ HCF = 2^2 \times 3^1 = 4 \times 3 = 12 ]
Method 2: Division Method
The division method can also be used, where you divide the larger number by the smaller number and continue with the remainder until you reach zero:
- Divide 24 by 12:
- 24 ÷ 12 = 2 (remainder 0)
Since the remainder is zero, the last divisor, 12, is the HCF.
Summary of HCF of 12 and 24
To summarize the findings:
- The factors of 12 are 1, 2, 3, 4, 6, 12.
- The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
- The common factors are 1, 2, 3, 4, 6, 12.
- Therefore, the Highest Common Factor of 12 and 24 is 12.
Important Notes
- “The HCF is useful in simplifying fractions, finding equivalent fractions, and solving problems involving ratios.”
- The HCF can also be used in various applications, including arithmetic operations, geometry, and algebra.
Final Thoughts
Understanding how to calculate the HCF is essential for many mathematical applications, including simplifying ratios and fractions. By utilizing methods like listing factors, prime factorization, and the division method, you can easily find the HCF of any two numbers. In this case, we learned that the HCF of 12 and 24 is 12, demonstrating that sometimes the numbers themselves can be their greatest common factor.